Sadly, there are many Christians and people claiming to be Christians nowadays who are totally confused about what the Trinity teaches and what the Trinity does not teach. I have seen Modalists in the blogosphere who not only state Trinitarians are pagans and tritheists but completely misinterpret and abuse scripture to arrive at the erroneous conclusion that Jesus is the only person of God.
So what precisely does Trinitarian theology teach? Well a good starting place can be found in the Athanasian Creed.
we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons; nor dividing the Essence. For there is one Person of the Father; another of the Son; and another of the Holy Ghost. But the Godhead of the Father, of the Son, and of the Holy Ghost, is all one; the Glory equal, the Majesty coeternal. Such as the Father is; such is the Son; and such is the Holy Ghost. The Father uncreated; the Son uncreated; and the Holy Ghost uncreated. The Father unlimited; the Son unlimited; and the Holy Ghost unlimited. The Father eternal; the Son eternal; and the Holy Ghost eternal. And yet they are not three eternals; but one eternal. As also there are not three uncreated; nor three infinites, but one uncreated; and one infinite.
And so there we see summed up how God is one in the context and understanding of the oneness of God in relation to Deuteronomy 6:4. The anti-Trinitarians highly praise this verse as their anthem of attack against the Trinity but the scriptures do not define how God is one nor does it explain “unitarianism” in the Bible. They are one infinite, one Almighty, one uncreated.
As for the term “persons”, St Augustine makes certain to point that these are helpful identifiers and means to describe the relationship of the three but this language is overall inadequate. He writes in On the Holy Trinity in book 5,
Chapter 9.— The Three Persons Not Properly So Called [in a Human Sense].
But because with us the usage has already obtained, that by essence we understand the same thing which is understood by substance; we do not dare to say one essence, three substances, but one essence or substance and three persons: as many writers in Latin, who treat of these things, and are of authority, have said, in that they could not find any other more suitable way by which to enunciate in words that which they understood without words. For, in truth, as the Father is not the Son, and the Son is not the Father, and that Holy Spirit who is also called the gift of God is neither the Father nor the Son, certainly they are three. And so it is said plurally, I and my Father are one. For He has not said, is one, as the Sabellians say; but, are one. Yet, when the question is asked, What three? human language labors altogether under great poverty of speech. The answer, however, is given, three persons, not that it might be [completely] spoken, but that it might not be left [wholly] unspoken.
Thus, we see in the Trinity a bit of an apophatic theology. We can grasp it when we understand what it is not but we cannot explain it exactly in the way it is. This applies also to the term “persons”. I have come across a Modalist who stated that Trinitarians believe the Father, Son, and Holy Spirit were actually three Spirits but this is not the case. The three are all in fact one Spirit. This is the mystery of the Trinity. The terms “Father, Son, and Holy Spirit” are not merely titles or hats God wears as revealed to us by scriptures but they are in fact the names in how each one relates to each other within the Trinity. Within the Trinity is important. The Trinity actually does not deny the fatherhood of Christ!
So the Trinity teaches the following propositions–the Father is God, the Son is God, the Holy Spirit is God. They are one in essence and not each other. Do the scriptures coincide?
The Father is God (John 17:3)–I do not think we need any more proof about this as this is universally agreed.
The Son is God (John 1:1, claims to be God in John 14:6-11 (Arians typically state that Jesus was validated being the Messiah to say he was the Truth but this is still equating himself to deity and would be incredibly blasphemous for him to see if one saw him, one saw the Father if he was not God), Titus 2:13, many others).
The Holy Spirit is also identified as God in Acts 5:1-4 (can be lied to, some say this is just an impersonal force but a force cannot be lied to), more soever the Holy Spirit is blasphemed and this is the deadliest of sin of all (Matt. 12:31-32). If the Holy Spirit is not God or a person of God, then these texts make no sense.
They are not each other. We see that scriptures reveal to us in Matt. 3:16-17 God declaring Jesus his Son and the Holy Spirit descending. Jesus is not saying that he is his own Son as the Modalists say. We also see greetings from all of them in 2 Cor. 13:13-14 and we see all three of them together in Rev. 4-5.
That was a rapid-fire sermonette on the Trinity in the scriptures. There are many more too and it requires exegetical gymnastics to refute these texts. It is in the life of the Church though ultimately that the Trinity becomes a reality and only in participating in the life of the Church can one in error about the Trinity ever hope to be corrected about it.