Remember the important principle that the Bible interprets the Bible. What this particular passage shows us is that the process of baptism and entering God's family involves the Father, the Son and the Holy Spirit. It is not a description of the nature of God.

Notice Acts 2:38 Acts 2:38Then Peter said to them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and you shall receive the gift of the Holy Ghost.
American King James Version×: "Then Peter said to them, 'Repent, and let every one of you be baptized in the name of Jesus Christ for the remission of sins; and you shall receive the gift of the Holy Spirit." After real repentance and being baptized, the example from Scripture is that a minister lays his hands on the repentant person and he or she receives the Holy Spirit directly from God (Acts 8:14-17 Acts 8:14-17 [14] Now when the apostles which were at Jerusalem heard that Samaria had received the word of God, they sent to them Peter and John: [15] Who, when they were come down, prayed for them, that they might receive the Holy Ghost: [16] (For as yet he was fallen on none of them: only they were baptized in the name of the Lord Jesus.) [17] Then laid they their hands on them, and they received the Holy Ghost.
American King James Version×).

Important as it is, baptism alone is not sufficient. We must receive God's Holy Spirit—the seed of eternal life—which is imparted by the biblically mandated laying on of hands of one of God's ministers (Acts 19:1-6 Acts 19:1-6 [1] And it came to pass, that, while Apollos was at Corinth, Paul having passed through the upper coasts came to Ephesus: and finding certain disciples, [2] He said to them, Have you received the Holy Ghost since you believed? And they said to him, We have not so much as heard whether there be any Holy Ghost. [3] And he said to them, To what then were you baptized? And they said, To John's baptism. [4] Then said Paul, John truly baptized with the baptism of repentance, saying to the people, that they should believe on him which should come after him, that is, on Christ Jesus. [5] When they heard this, they were baptized in the name of the Lord Jesus. [6] And when Paul had laid his hands on them, the Holy Ghost came on them; and they spoke with tongues, and prophesied.
American King James Version×). We cannot partake of God's nature, as mentioned in 2 Peter 1:4 2 Peter 1:4Whereby are given to us exceeding great and precious promises: that by these you might be partakers of the divine nature, having escaped the corruption that is in the world through lust.
American King James Version×, without first being begotten of the Father by the Holy Spirit, which imparts that divine nature.

Why are Father, Son and Holy Spirit mentioned in Matthew 28:19 Matthew 28:19Go you therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:
American King James Version×? Let us understand. At baptism we enter into a covenant relationship with God the Father. The sacrifice of Jesus Christ, the Son of God, makes that covenant relationship possible. (Of course, through our repentance and baptism we also enter into a relationship with Jesus Christ as our older Brother and head of the Church.) The Holy Spirit is the means by which the Father and Son make all of this possible.

On another level, God the Father is the One who calls us to baptism and a new way of life (John 6:44 John 6:44No man can come to me, except the Father which has sent me draw him: and I will raise him up at the last day.
American King James Version×, John 6:65 John 6:65And he said, Therefore said I to you, that no man can come to me, except it were given to him of my Father.
American King James Version×), and it is His goodness that leads us to repentance and baptism (Romans 2:4 Romans 2:4Or despise you the riches of his goodness and forbearance and long-suffering; not knowing that the goodness of God leads you to repentance?
American King James Version×). We know also that Jesus Christ died as a sacrifice for our sins, reconciling us to God (Romans 5:6-11 Romans 5:6-11 [6] For when we were yet without strength, in due time Christ died for the ungodly. [7] For scarcely for a righteous man will one die: yet peradventure for a good man some would even dare to die. [8] But God commends his love toward us, in that, while we were yet sinners, Christ died for us. [9] Much more then, being now justified by his blood, we shall be saved from wrath through him. [10] For if, when we were enemies, we were reconciled to God by the death of his Son, much more, being reconciled, we shall be saved by his life. [11] And not only so, but we also joy in God through our Lord Jesus Christ, by whom we have now received the atonement.
American King James Version×)—baptism picturing our burial with Jesus Christ and our being raised to a new life with Him both now and in the future resurrection (Romans 6:1-5 Romans 6:1-5 [1] What shall we say then? Shall we continue in sin, that grace may abound? [2] God forbid. How shall we, that are dead to sin, live any longer therein? [3] Know you not, that so many of us as were baptized into Jesus Christ were baptized into his death? [4] Therefore we are buried with him by baptism into death: that like as Christ was raised up from the dead by the glory of the Father, even so we also should walk in newness of life. [5] For if we have been planted together in the likeness of his death, we shall be also in the likeness of his resurrection:
American King James Version×). And God's Holy Spirit, as we will see more in a later chapter, is what makes us God's begotten children (Romans 8:16 Romans 8:16The Spirit itself bears witness with our spirit, that we are the children of God:
American King James Version×).

The instruction in Matthew 28:19 Matthew 28:19Go you therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:
American King James Version×presumes that, before being baptized, believers will learn of God the Father, His Son Jesus Christ and the Holy Spirit. At baptism, these believers enter into a personal family relationship with God the Father and the Son through the Holy Spirit, thereby receiving the name of God (compare Ephesians 3:14-15 Ephesians 3:14-15 [14] For this cause I bow my knees to the Father of our Lord Jesus Christ, [15] Of whom the whole family in heaven and earth is named,
American King James Version×).

Note again that Father, Son and Holy Spirit are all acknowledged as being involved in this process. But that does not mean all three are divine persons in a Trinity. To claim that Matthew 28:19 Matthew 28:19Go you therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:
American King James Version×establishes one God in three persons goes far beyond the actual words of the verse. And other verses show such a notion to be utterly false.