That's a great question. I'll give you a couple different angles to look at this from. First, we could argue that the "wise men" who visit Jesus in His infancy are the first believers. Even though they might not have known exactly what the significance of Jesus' life is, it seems they had some kind of special knowledge about the child Jesus, as they traveled an extraordinarily long way to see Him, and when they arrive they say "Where is the one who has been born king of the Jews?" We know, by this, that they were at least somewhat aware of Jesus' significance. So you could say they were the first believers, because it's possible they were already on their journey before Jesus was anywhere close to being born (Reference Matthew 2).
Okay, here's the second candidate: Simeon. Simeon was a man we don't know a lot about, except that he truly loved the God of
Israel and that it had been promised to him that he would see the Messiah before he died. Sure enough, Jesus' parents brought Him to the temple (this was a tradition that they would take babies to the temple to "dedicate" them to the Lord) and Simeon blesses Jesus, acknowledging belief in His holiness. So Simeon definitely recognized Jesus as true Messiah, even though Jesus was only a baby (Reference Luke 2).
John the Baptist is our third choice. Like the wise men and Simeon, he has a bit of an advantage because he had special guidance from God that let him know who Jesus was. He started preaching that the Messiah was coming before Jesus was well-known and when Jesus shows up one day where John was preaching, he calls out in front of everyone that Jesus is the Messiah that's been promised (John 1:21-34). We know that John truly believed in Jesus even before they met.
Okay, last example: in John 1, we see that soon after John proclaims to everyone that Jesus is the Messiah, John was with a few of the men that would hang out with him and listen to his preaching (the scriptures calls them "John's disciples"). They look up and see Jesus passing by and John yells out "Look! the Lamb of God!" When he says this, those two disciples of John immediately go and follow Jesus and, after spending a little time with Him, one of the men named "Andrew" goes and gets his brother Peter and says "We have found the Messiah." From this we know that Andrew immediately believed Jesus was the promised one, the savior of Israel, and he was probably the first person who believed because of someone's testimony alone. He didn't have God coming to him with an angel or vision to tell him Jesus was the Messiah, he just heard John say it and he believed. Mark's gospel tells the story a little differently, saying that Peter and Andrew were fishing together when Jesus was passing by, however the facts of these stories could well be true of the same event (Reference John 1 and Mark 1 for comparison).