The Trinity is not really mentioned in the Old Testament. The angel of the Lord is just that--an angel. The angel of the Lord is a special representative or messenger of God to God's people, and according to the ancient concept of agency, he could be considered to be the Lord who sent them, and was to be treated as if he were the one who sent him.
According to Christian doctrine, there was no incarnation of Jesus prior to the Incarnation. This is not to say that the divine son of God was not involved in creation and other things prior to his taking on flesh in Mary's womb; it is simply to make clear that he was no mere angel of the Lord, nor did he manifest himself in some observable form prior to the Incarnation.
Phillippians 2.5-11 should be compared with John 1. Both these texts make clear that the divine Son of God existed before he took on flesh. Indeed, John 1 makes clear he existed before the creation of the universe, and even helped in the
creation of the universe. Thus, John 1 makes clear that there are at least two persons in the Godhead. Since elsewhere in the New Testament (see, for example, John 14-17), the Holy Spirit is treated as a person distinct from the Father and the Son, not merely as a force or power of God, there is an implicit assumption that the Spirit is also a person within the Godhead. In Matthew 28, the fact that baptism is done in the name of Father, Son, and Spirit is important for baptism would only be done in God's name.
Notice that Matthew 28.19 does not speak of names plural, but of the name of God (singular) being Father, Son, and Spirit--three persons, one Godhead.
Finally, there is a functional subordination of the Son to the Father, and the Spirit to both the Son and the Father, that leads to statements like we find in John's gospel, where Jesus indicates that he must wait on the leading and guidance of God before acting. The Son acts for the Father and on the basis of the Father's guidance, and the spirit (called another advocate in John 14-17) acts for the Son. There is thus equality of nature and being between Father, Son and Spirit, but functional subordination in terms of how the Trinity works.