Part I: Definition and Historical Background, Relying On Others To Decide What We Should Believe
Part II - Plurality in The Godhead
Part III - The Deity of Christ & The Deity of The Holy Spirit
You Are Here Part IV: Passages That Supposedly "Prove" the Trinity.
Part V: The Grammar... Can it legitimately be used to support the idea that the Holy Spirit is the third Person of the Trinity
Part VI: The Holy Spirit... a Separate Person, Or The Divine Presence And Power Of The Father Himself
Part VII: The doctrine of the trinity that has remained virtually unchanged to this day found its roots in paganism not the Bible. This largely due to the part played by the Cappadocian Fathers - three ancient Greek philosophers and mystics.
Part VIII: The Son is "begotten" of the Father, and the Spirit "proceeds" from the Father.
A difference yes, but not what The Cappadocian Fathers made it out to be
Part IX: Summary and Conclusion
Asking a person to read the actual text, and not allow anyone to tell them what they think the text means, seems to be quite a novel concept in Christianity. When someone who already believes in the trinity reads any Biblical passages that mentions Father, Son, and Holy Spirit in one breath - they immediately impose the concept of the trinity onto those texts and go "Aha! Here's the proof"
Are we just so used to having the Bible interpreted for us that we are too lazy, or too gullible to go look for ourselves? Or have we been so indoctrinated that when we read the verses in question, we read into them what we have been led to believe they say / have always been told what their meaning is. Or do we, as a matter of course, just believe the man behind the pulpit, the one with a string of letters behind his name, or the one who's words happened to find themselves in print (After all, he must know what he is talking about).
It is not surprising that, over and over again, Jesus called us sheep.
If we, without prejudice, or preconceived ideas stick to exactly what the text says and abstain from going beyond the actual words, we will find that none of the verses usually cited say anything about the nature of the Father, Son and Holy Spirit. None of them describe a triune Godhead, or even imply that there are three coequal Persons in one Divine Being.
'Proof Texts' For The Holy Spirit Being a "Person" in His Own Right
A number of Biblical texts that mention Father, Son and Holy Spirit in one place are often pointed to as 'proof texts' for the Trinity.
Kenneth Samples, research scholar at Reasons To Believe, says "six simple statements show how the doctrine of the trinity "is indeed derived from Scripture". The first four statements refer to the fact that there is only one true God, and the Father, Son and Holy Spirit are all "called or referred to as God".
No argument there. They are.
The Holy Spirit - A Distinct Person?
The fifth statement is "The Father, Son, and Holy Spirit are distinct persons and can be distinguished from one another (the Father is not the Son; the Father is not the Holy Spirit; and the Son is not the Holy Spirit.  In support of this statement he refers to the following verses...
1. Go therefore and make disciples of all the nations, baptizing them in the name of the Father and the Son and the Holy Spirit, (Matthew 28:19 NASB) (More about this verse below)
2. and the Holy Spirit descended upon Him in bodily form like a dove, and a voice came out of heaven, "You are My beloved Son, in You I am well-pleased." (Luke 3:22 NASB)
3. When the Helper comes, whom I will send to you from the Father, that is the Spirit of truth who proceeds from the Father, He will testify about Me, (John 15:26 NASB)
4. But when He, the Spirit of truth, comes, He will guide you into all the truth; for He will not speak on His own initiative, but whatever He hears, He will speak; and He will disclose to you what is to come. "He will glorify Me, for He will take of Mine and will disclose it to you. "All things that the Father has are Mine; therefore I said that He takes of Mine and will disclose it to you. (John 16:13-15 NASB)
5. The grace of the Lord Jesus Christ, and the love of God, and the fellowship of the Holy Spirit, be with you all. (2 Corinthians 13:14 NASB)
However, If you read what the text says without superimposing anyone's interpretation onto it you should realize that
Quote 1 could be used to support both Tritheism (the belief that the Father, the Son, and the Holy Spirit are three separate Gods) and Modalism that (the belief that although God is a single Being He has, at different times, worn different hats and revealed Himself in different forms - the Father in Old Testament times, the Son in the New and the Holy Spirit after the resurrection).
Quote 2 does nothing but mention Father, Son and Holy Spirit in one place.
In fact, none of the verses quoted say a single word about the nature of the Holy Sprit nor about unity and/or equality. All of these ideas have to be read into the text.
A Triadic Pattern of Unity And Equality?
Mr. Samples also states "The three persons (Father or God; and Son or Christ or Lord; and Holy Spirit or Spirit) are frequently listed together in a triadic pattern of unity and equality (Romans 15:16, 30; 1 Corinthians 12:4-6; 2 Corinthians 1:21–22; Galatians 4:6)." , which is a classic case of reading far more into the verses than the text actually says.
I am not going to list the verse here but again, if you read exactly what the text says not what you have been told it means, you should realize that these verses do nothing but show Father, Son, and Holy Spirit exist. There isn't a single verse in the New Testament that says anything about unity and/or equality, much less about the nature of the Holy Sprit, or the nature of Jesus Himself. In fact, the Oxford Companion to the Bible says
While the New Testament writers say a great deal about God, Jesus and the Spirit of each, no New Testament writer expounds on the relationship among the three in the detail that later Christian writers do. 
Since Father, Son, and Holy Spirit are all equally involved in the process of human salvation, it is little wonder that they are mentioned together in some contexts but, unless we read our own ideas into the these verses, none of them, in any way, substantiate the orthodox view of the Trinity as God in three Persons.
One verse often cited as 'proof' is...
1 John 5:7
1 John 5:7 is often pointed to as proof of the three Persons in Heaven. However it is widely believed that this short clause was a later addition. The King James version reads (the disputed verse has been underlined)
(5) Who is he that overcometh the world, but he that believeth that Jesus is the Son of God? (6) This is he that came by water and blood, even Jesus Christ; not by water only, but by water and blood. And it is the Spirit that beareth witness, because the Spirit is truth. (7) For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one. (8) And there are three that bear witness in earth, the Spirit, and the water, and the blood: and these three agree in one. (1 John 5:5-8 KJV)
This controversial portion of 1 John 5 is known as the Comma Johanneum, which is a comma or short clause. Because this sentence provides an explicit reference to the doctrine that the Father, Son and Holy Spirit are one God, some Christians are resistant to the elimination of the Comma from modern Biblical translations.
Nonetheless, most scholars now believe that the theology contained in the Comma is true, but that the Comma itself was not in John's original Epistle but was added to provide unambiguous evidence for the doctrine. Both the internal and external evidence speak against the authenticity of the passage.
The Comma Johanneum isn't found in any early Greek manuscripts, nor in early translations. As explained by Wayne Jackson of Christian Courier (Emphasis added)
The passage is found in only four Greek manuscripts (of more than 5,000 available ones), none of which dates before the eleventh century A.D. Even in these manuscripts, it appears that the passage has been rendered from a late edition of the Latin Vulgate....
The passage is absent from the manuscripts of all the ancient versions into which the Greek had been translated, e.g., Syriac, Coptic, Ethiopic, Arabic, and the Old Latin, and the Vulgate (in their early forms).
The earliest instance of this phraseology is found in a 4th century essay titled, Liber Apologeticus. From thence it found its way into the writings of the Latin Fathers, and into the Old Latin and Vulgate versions (c. 5th and 8th centuries respectively).
So why was it was present in most translations of 1 John published from 1522 until the latter part of the nineteenth century? The answer is simply because the third edition of the Textus Receptus (or received version) was the sole source for translation. As Wayne Jackson goes on to say
"When Erasmus published the early editions of his Greek New Testament (1516, 1519), he was criticized for not including the spurious sentence. Yielding to pressure, he promised to put it in a later edition if it could be found in only one manuscript. Subsequently, a copy was produced — apparently made to order! — and Erasmus incorporated it into his third edition (1522). From there it made its way into the Textus Receptus (the so-called Received Text) and finally into the King James Version". 
The Big Book of Bible Difficulties tells us:
"This verse has virtually no support among the early Greek manuscripts . . . Its appearance in late Greek manuscripts is based on the fact that Erasmus was placed under ecclesiastical pressure to include it in his Greek NT of 1522, having omitted it in his two earlier editions of 1516 and 1519 because he could not find any Greek manuscripts which contained it" .
This is widely supported by other Biblical scholars who now largely agree that the theology contained in the Comma is true, but that the Comma is not an original part of the Epistle of John. The textual evidence is against 1 John 5:7 explains Dr. Neil Lightfoot, a New Testament professor (Emphasis Added)
Of all the Greek manuscripts, only two contain it. These two manuscripts are of very late dates, one from the fourteenth or fifteenth century and the other from the sixteenth century. Two other manuscripts have this verse written in the margin. All four manuscripts show that this verse was apparently translated from a late form of the Latin Vulgate" 
Theology professors Anthony and Richard Hanson, in their book Reasonable Belief: A Survey of the Christian Faith, explain the unwarranted addition to the text this way: (Emphasis Added)
"This was a late interpolation quite certainly absent from the original text. It was added by some enterprising person or persons in the ancient Church who felt that the New Testament was sadly deficient in direct witness to the kind of doctrine of the Trinity which he favoured and who determined to remedy that defect . . . It is a waste of time to attempt to read Trinitarian doctrine directly off the pages of the New Testament" 
Nearly all recent translations have removed this clause. However, there are still some who cry "foul", to which bible.org says
That KJV advocates have charged modern translations with heresy because they lack the Comma is a house of cards, for the same translators who have worked on the NIV, NASB, or NET (as well as many other translations) have written several articles and books affirming the Trinity. 
Several early sources which one might expect to include the Comma Johanneum, have not.
Although Clement of Alexandria's writings around the year 200 place a strong emphasis on the Trinity, his quotation of 1 John 5:8 does not include the Comma. In fact, none of the early Church Fathers quoted this part of the passage, in spite of the fact that this verse (especially the latter half which states that these three are one) would have greatly contributed to the strength of their arguments in, for example, their Trinitarian debates with the Arians.
Why? Simply because this clause did not exist at the time.
In any case, the addition says nothing about the nature of the Father, Word and Holy Spirit, but merely lists all three as witnesses in heaven. Note that verse 8 lists three witness on earth (the Spirit, and the water, and the blood), two of which are inanimate objects.
Other Passages That Supposedly "Prove" the Trinity
The following "proof texts" merely state that Father, Son and Holy Spirit exist. All are very important, and all are, in some way, involved in whatever aspect of the Christian spiritual life is being described or discussed. Additionally, every single verse holds true if, as shown in the next chapter, the Holy Spirit is the Spirit of the Father Himself. A perfect example of this is...
After being baptized, Jesus came up immediately from the water; and behold, the heavens were opened, and he saw the Spirit of God descending as a dove and lighting on Him, and behold, a voice out of the heavens said, "This is My beloved Son, in whom I am well-pleased." (NASB)
This verse is generally supposed to indicate three co-equal member of the Godhead. However, anyone who doesn't read meaning into the text, but simply reads the text itself, will realize that all this verse tells us is what the Father, Son and Holy Spirit did on this occasion... Jesus was baptized, the Spirit descended on Him, and the Father spoke from Heaven. There is not even the slightest hint that the Spirit is a person, much less that there are three Persons in one being.
Perhaps, this can be better understood by an analogy
Unless you happen to know who exactly I was speaking about, you would not be any the wiser were I to tell you that Father, Sam and Charlie are on the porch. While you might assume I am referring to my father, is Sam referring to Samuel the next door neighbor, Samantha my cousin, or someone else? And you couldn't possibly know whether Charlie is human, my Labrador, or pet canary.
Similarly, if Matthew 3:16-17 were the only verse you ever read from the Bible, you wouldn't have the faintest clue as to what Father and Son was being referred to. And you would really be scratching your head at the mention on a "Holy Sprit". The point being that just as you needed prior knowledge to know just who is on the porch, you need independent knowledge to know who Matthew was referring to. Just as your prior knowledge of Father and Son came from other parts of the Bible, so too whatever knowledge we have regarding the Holy Spirit also has to come from other parts of the Bible.
From our perspective any three beings/persons that perform different actions simultaneously must mean each of the three has to be independent of each other.
Therefore, being forced to reconcile our understanding of three independent Beings with the fact that God says He is one, has led to formulate a totally incomprehensible explanation about God being three in Person, but "one in essence". This allows us the luxury of thinking that our doctrine in no way contradicts the famous Shema Yisrael, when Moses said "Hear, O Israel! The Lord is our God, the Lord is one (Heb. echâd)! (Deuteronomy 6:4)
Nice, tidy solution! Problem solved! Case closed!
Except the problem did not exist in the first place, simply because the Hebrew echâd can also refer to a collective unit. See Chapter 3
is often assumed to be a proof text for the orthodox view of one God in three persons, ie. Father, Son, and Holy Spirit. One of the last things Jesus told His disciples was,
"Go therefore and make disciples of all the nations, baptizing them in the name of the Father and the Son and the Holy Spirit, teaching them to observe all that I commanded you; and lo, I am with you always, even to the end of the age." (Matthew 28:19-20 NASB
However, if we just read His words and abstain from reading our own ideas into them it is clear that the text itself is not a description of the nature of God. Jesus did not tell us who they are much less what their exact relationship is.
Think about it for a moment - the text could be used to support Tritheism that teaches that the Father, the Son, and the Holy Spirit are three separate Gods and Modalism that teaches that God although a single Being has, at different times, worn different hats and revealed Himself in different forms (the Father in Old Testament times, the Son in the New and the Holy Spirit after the resurrection) but never at the same time.
So why did Jesus instruct His disciples to baptize converts in the name of the Father and the Son and the Holy Spirit, if not to show a Triune God? To begin with there was a distinct difference between John's baptism and the one initiated by Jesus. Because the Spirit had not yet been given John the Baptist's baptism for repentance was with water only. As he said, "... He who is coming after me is mightier than I, and I am not fit to remove His sandals; He will baptize you with the Holy Spirit and fire. (Matthew 3:11 NASB).
While there is no record of Jesus personally baptizing anyone, the disciples did so in His name. However, they did not start baptizing until after Pentecost which is when the Spirit came down on men. At baptism we enter into a covenant relationship with God the Father, only made possible through the sacrifice of Jesus Christ. However, receiving the Spirit is an integral and indispensable part of becoming a Christian.
It happened that while Apollos was at Corinth, Paul passed through the upper country and came to Ephesus, and found some disciples. He said to them, "Did you receive the Holy Spirit when you believed?" And they said to him, "No, we have not even heard whether there is a Holy Spirit." And he said, "Into what then were you baptized?" And they said, "Into John's baptism." Paul said, "John baptized with the baptism of repentance, telling the people to believe in Him who was coming after him, that is, in Jesus." When they heard this, they were baptized in the name of the Lord Jesus. And when Paul had laid his hands upon them, the Holy Spirit came on them, and they began speaking with tongues and prophesying. (Acts 19:1-6 NASB)
The Spirit was given us as a pledge (2 Corinthians 1:22)
if anyone does not have the Spirit of Christ, he does not belong to Him" (Romans 8:9).
For all who are being led by the Spirit of God, these are sons of God. For you have not received a spirit of slavery leading to fear again, but you have received a spirit of adoption as sons by which we cry out, "Abba! Father!" The Spirit Himself testifies with our spirit that we are children of God, (Romans 8:14-16 NASB)
Matthew 28:19 - A Baptismal Formula?
Melinda Penner says (Emphasis Added)
Admission to the church was connected with belief in the doctrine of the Trinity as evidenced by the early baptismal formula that was used in accordance with the command of Jesus in Matthew 28:19. The doctrine of the Trinity was essential from the beginning of the Christian Church. 
Because if Jesus' command was that this verbal formula be used on every occasion, Peter really messed up when, on Pentecost, he told the Jews ... (Emphasis Added)
"Repent, and each of you be baptized in the name of Jesus Christ for the forgiveness of your sins; and you will receive the gift of the Holy Spirit. (Acts 2:38 NASB)
Did Peter forget the words the Lord had told him to use, decide to ignore the Holy Spirit, or did he simply understand, as we seem not to, that the very act of baptism involves Father, Son and Holy Spirit.
In other words, when Jesus told His disciples to baptize in the name of the Father, Son and Holy Spirit, it was an acknowledgment that all three are involved in the process, and not intended to be a precise formula used as admission to the church, nor support for the orthodox version of the Trinity.
Several verses in the epistles are used to support belief in a Trinity. However, none of them do anything but show that Father, Son and Holy Spirit all exist and are all necessary to both become and remain a Christian. None of them speak of, or even hint at a triune God. And, even more strongly, none of them present the Spirit as a separate being. Once again, read what the text says, not what you have been told the text means.
2 Corinthians 13:14: The grace of the Lord Jesus Christ, and the love of God, and the fellowship (Gr. koinonia) of the Holy Spirit, be with you all. (NASB
This is the only place in Paul's writings that he mentions the Holy Spirit in the same verse as the Father and Son. However, Paul is not making a theological statement about the nature of God, but merely says God's spirit is the unifying agent that bring us together, presumably with the Father and Son and with each other. What is to be particularly taken note of is the fact that God's Spirit is not spoken of as a person, nor does Paul say our fellowship is with the Holy Spirit.
In fact, in his epistles, John speaks about fellowship with the Father, the Son, and with each other, but leaves the Holy Spirit out .
"what we have seen and heard we proclaim to you also, so that you too may have fellowship with us; and indeed our fellowship is with the Father, and with His Son Jesus Christ. (1 John 1:3 NASB)
So what did Paul mean when he talked about the fellowship of the Holy Spirit. The Greek word translated fellowship is koinonia, which Paul also used in His letter to the Philippians.
Therefore if there is any encouragement in Christ, if there is any consolation of love, if there is any fellowship (Gk. koinonia) of the Spirit, if any affection and compassion, make my joy complete by being of the same mind, maintaining the same love, united in spirit, intent on one purpose. (Philippians 2:1-2 NASB)
Over and over Paul stressed the unity of believers. He exhorts them to be "like minded" (2 Corinthians 13:11), walk in love (Ephesians 5:2), stand firm in one spirit, and with one mind strive together for the faith of the gospel (Philippians 1:27). This was all accomplished because they all have the same unifying Spirit dwelling within them.
"There is one body and one Spirit, just as also you were called in one hope of your calling; one Lord, one faith, one baptism, one God and Father of all who is over all and through all and in all. (Ephesians 4:4–6 NASB)
For by one Spirit we were all baptized into one body, whether Jews or Greeks, whether slaves or free, and we were all made to drink of one Spirit. (1 Corinthians 12:13 NASB)
Paul was simply concluding his epistle with an exhortation of unity through the Holy Spirit. This is almost exactly the same concept expressed in...
Ephesians 2:18: for through Him we both have our access in one Spirit to the Father. So then you are no longer strangers and aliens, but you are fellow citizens with the saints, and are of God's household, (NASB)
Any trinitarian implications of this verse are only obvious to those with a pre-bias. Just as Paul exhorted the Corinthians to unity through the Holy Spirit (above), he told the Ephesians that "There is one body and one Spirit" (4:4–6) and through Christ we have access to the Father through this one Spirit. The verse says nothing of the Spirit being a Person.
Romans 15:30: I know that when I come to you, I will come in the fullness of the blessing of Christ. Now I urge you, brethren, by our Lord Jesus Christ and by the love of the Spirit, to strive together with me in your prayers to God for me. (NASB)
The "love of the Spirit" is the love of God that is put in our hearts through the Holy Spirit (Romans 5:5). In other words, the Spirit of God produces love and sympathy in our hearts. Paul asks the Galatians to manifest that love by praying earnestly for him. There is absolutely nothing in this verse that says, or even implies, that the Spirit is a person.
Again, Romans 15:30 simply shows that Jesus, the Spirit and God the Father exist, not that they form a triune being.
1 Peter 1:2: according to the foreknowledge of God the Father, by the sanctifying work of the Spirit, to obey Jesus Christ and be sprinkled with His blood: May grace and peace be yours in the fullest measure. (NASB)
In context this verse says the Holy Spirit sanctifies us to obey Christ. However, it does not say nor imply that the Holy Spirit is a separate being. In any case the work of sanctification is not limited to the Sprit alone...
Now may the God of peace Himself sanctify you entirely; and may your spirit and soul and body be preserved complete, without blame at the coming of our Lord Jesus Christ. (1 Thessalonians 5:23 NASB)
Therefore Jesus also, that He might sanctify the people through His own blood, suffered outside the gate. (Hebrews 13:12 NASB)
In order to come to a sound conclusion about the trinity (or, for that matter, any other Biblical doctrine), one has to do some digging instead of coming to conclusions based on a superficial reading of the text and not taking other passages into consideration. If we are willing to invest our time and apply our minds, our shovels will uncover much that contradicts the orthodox view.
Continue On To Part VI - The Grammar. New Testament grammar is often used by many evangelicals as the first line of defense against any challenges to the doctrine. The problem is that the grammar cannot necessarily be used to support the idea that the Holy Spirit is a "he", much less the third Person of the Trinity. Doctrinal bias, not grammatical accuracy, is responsible for referring to the Holy Spirit with masculine rather than neuter pronouns when both are equally legitimate. HERE
 Kenneth Samples. The Trinity's Biblical Basis. http://reflectionsbyken.wordpress.com/2011/08/16/the-trinity’s-biblical-basis/
 The Oxford Companion to the Bible. Editors Bruce M. Metzger and Michael David Coogan. Oxford University Press, USA; First Edition edition (October 14, 1993 Pg 782
 Wayne Jackson. Defending the Faith with a Broken Sword – Part 3.
 Norman Geisler and Thomas Howe, The Big Book of Bible Difficulties. Baker Books (June 1, 2008), Pgs. 540-541
 Dr. Neil Lightfoot. How We Got the Bible, Baker Books; 3 edition (June 1, 2010, Pgs. 100-101
 Anthony and Richard Hanson. Reasonable Belief A Survey of the Christian Faith. Oxford University Press (January 25, 1990), Pg. 171
 Daniel B. Wallace , Th.M., Ph.D. The Textual Problem in 1 John 5:7-8. https://bible.org/article/textual-problem-1-john-57-8
 Melinda Penner. The Doctrine of the Trinity at Nicaea and Chalcedon.