Why was Jesus baptized in the New Testament? Moreover, why does baptism even exist in the New Testament? After reading the old it seems it is not present anywhere.
John the Baptist was baptizing in repentance of sin:
1 In those days John the Baptist came preaching in the wilderness of Judea, 2 "Repent, for the kingdom of heaven is at hand." 3 For this is he who was spoken of by the prophet Isaiah when he said, "The voice of one crying in the wilderness: 'Prepare the way of the Lord; make his paths straight.'" - Matthew 3:3 - Matthew 3:1-3 ESV
Bear fruit in keeping with repentance. - Matthew 3:8 ESV
"I baptize you with water for repentance, but he who is coming after me is mightier than I, whose sandals I am not worthy to carry. He will baptize you with the Holy Spirit and fire. - Matthew 3:11 ESV
But Jesus was without sin, and not in need of repentance, a point that John himself observes:
13 Then Jesus came from Galilee to the Jordan to John, to be baptized by him. 14 John would have prevented him, saying, "I need to be baptized by you, and do you come to me?" - Matthew 3:13-14 ESV
But Jesus tells him that:
15 But Jesus answered him, "Let it be so now, for thus it is fitting for us to fulfill all righteousness." Then he consented. 16 And when Jesus was baptized, immediately he went up from the water, and behold, the heavens were opened to him, and he saw the Spirit of God descending like a dove and coming to rest on him; 17 and behold, a voice from heaven said, "This is my beloved Son, with whom I am well pleased." - Matthew 3:15-17 ESV
This then, is a fufillment of prophecy from Isaiah:
3 A voice cries: "In the wilderness prepare the way of the LORD; make straight in the desert a highway for our God. 4 Every valley shall be lifted up, and every mountain and hill be made low; the uneven ground shall become level, and the rough places a plain. 5 And the glory of the LORD shall be revealed, and all flesh shall see it together, for the mouth of the LORD has spoken." - Isaiah 40:3-5 ESV
So John was baptizing people in a symbolic ceremony of repentance before the lord. Jesus was baptized in a symbolic ceremony of the fufillment of prophecy and ordination from the Father, through John. In doing so, it establishes Jesus authority and ministry.
Secondarily, and like all of Jesus' actions, it gives us a model for what we should be doing on Earth.
18 And Jesus came and said to them, "All authority in heaven and on earth has been given to me. 19 Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, 20 teaching them to observe all that I have commanded you. And behold, I am with you always, to the end of the age." - Matthew 28:18-20 ESV
Most Protestant churches baptize today in a symbolic ceremony of the sins that are washed away through faith in Christ and as an outward public proclamation of the transformation happening inside the believer.