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: "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).
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). We cannot partake of God's nature, as mentioned in 2 Peter 1:4, 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? 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 6:65), and it is His goodness that leads us to repentance and baptism (Romans 2:4). We know also that Jesus Christ died as a sacrifice for our sins, reconciling us to God (Romans 5:6-11)—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). 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).
The instruction in Matthew 28:19 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).
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 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.