Matthew's claim is that the new covenant proclaimed by Jeremiah was being fulfilled and the Messianic age, the kingdom of heaven, was being realized through the community that arose among Jesus' followers after he was killed by the authorities. One of strongest voices, according to other sources, in this Messianic movement was Peter's, but you wouldn't know that from Matthew's gospel, if it was the only source you had.
This is the last passage where Matthew mentions Peter by name:
26:69 Now Peter was sitting outside in the courtyard. A servant-girl came to him and said, “You also were with Jesus the Galilean.” 70 But he denied it before all of them, saying, “I do not know what you are talking about.” 71 When he went out to the porch, another servant-girl saw him, and she said to the bystanders, “This man was with Jesus of Nazareth.” 72 Again he denied it with an oath, “I do not know the man.” 73 After a little while the bystanders came up and said to Peter, “Certainly you are also one of them, for your accent betrays you.” 74 Then he began to curse, and he swore an oath, “I do not know the man!” At that moment the cock crowed. 75 Then Peter remembered what Jesus had said: “Before the cock crows, you will deny me three times.” And he went out and wept bitterly.
Here are the other highlights of Peter's time with Jesus, according to Matthew:
Peter was one of the first disciples called by Jesus:
4:18 As he walked by the Sea of Galilee, he saw two brothers, Simon, who is called Peter, and Andrew his brother, casting a net into the sea—for they were fishermen. 19 And he said to them, “Follow me, and I will make you fish for people.”
Peter was married:
8:14 When Jesus entered Peter’s house, he saw his mother-in-law lying in bed with a fever; 15 he touched her hand, and the fever left her, and she got up and began to serve him.
Peter had a tendency to overestimate his own faith, such as when Jesus walked on water:
14:28 Peter answered him, “Lord, if it is you, command me to come to you on the water.” 29 He said, “Come.” So Peter got out of the boat, started walking on the water, and came toward Jesus. 30 But when he noticed the strong wind,[e]he became frightened, and beginning to sink, he cried out, “Lord, save me!” 31 Jesus immediately reached out his hand and caught him, saying to him, “You of little faith, why did you doubt?” 32 When they got into the boat, the wind ceased. 33 And those in the boat worshiped him, saying, “Truly you are the Son of God.”
Mostly on the basis of this passage, Peter is called the first Pope:
16:13 Now when Jesus came into the district of Caesarea Philippi, he asked his disciples, “Who do people say that the Son of Man is?” 14 And they said, “Some say John the Baptist, but others Elijah, and still others Jeremiah or one of the prophets.” 15 He said to them, “But who do you say that I am?” 16 Simon Peter answered, “You are the Messiah,[c]the Son of the living God.” 17 And Jesus answered him, “Blessed are you, Simon son of Jonah! For flesh and blood has not revealed this to you, but my Father in heaven. 18 And I tell you, you are Peter,[d] and on this rock[e] I will build my church, and the gates of Hades will not prevail against it. 19 I will give you the keys of the kingdom of heaven, and whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.”20 Then he sternly ordered the disciples not to tell anyone that he was the Messiah.
But Jesus also called him Satan:
16:21 From that time on, Jesus began to show his disciples that he must go to Jerusalem and undergo great suffering at the hands of the elders and chief priests and scribes, and be killed, and on the third day be raised. 22 And Peter took him aside and began to rebuke him, saying, “God forbid it, Lord! This must never happen to you.” 23 But he turned and said to Peter, “Get behind me, Satan! You are a stumbling block to me; for you are setting your mind not on divine things but on human things.”
Peter and the sons of thunder were the inner circle:
17:1 Six days later, Jesus took with him Peter and James and his brother John and led them up a high mountain, by themselves. 2 And he was transfigured before them, and his face shone like the sun, and his clothes became dazzling white. 3 Suddenly there appeared to them Moses and Elijah, talking with him. 4 Then Peter said to Jesus, “Lord, it is good for us to be here; if you wish, I[a] will make three dwellings[b] here, one for you, one for Moses, and one for Elijah.”5 While he was still speaking, suddenly a bright cloud overshadowed them, and from the cloud a voice said, “This is my Son, the Beloved;[c] with him I am well pleased; listen to him!” 6 When the disciples heard this, they fell to the ground and were overcome by fear. 7 But Jesus came and touched them, saying, “Get up and do not be afraid.” 8 And when they looked up, they saw no one except Jesus himself alone.
Peter wanted to know just how often he had to forgive:
18:21 Then Peter came and said to him, “Lord, if another member of the church[g] sins against me, how often should I forgive? As many as seven times?” 22 Jesus said to him, “Not seven times, but, I tell you, seventy-seven[h] times.
Peter wanted to know what the payoff of his discipleship would be:
19:27 Then Peter said in reply, “Look, we have left everything and followed you. What then will we have?” 28 Jesus said to them, “Truly I tell you, at the renewal of all things, when the Son of Man is seated on the throne of his glory, you who have followed me will also sit on twelve thrones, judging the twelve tribes of Israel. 29 And everyone who has left houses or brothers or sisters or father or mother or children or fields, for my name’s sake, will receive a hundredfold,[d]and will inherit eternal life. 30 But many who are first will be last, and the last will be first.
I forgot the passage about the temple tax.
17: 24 When they reached Capernaum, the collectors of the temple tax came to Peter and said, “Does your teacher not pay the temple tax?” 25 He said, “Yes, he does.” And when he came home, Jesus spoke of it first, asking, “What do you think, Simon? From whom do kings of the earth take toll or tribute? From their children or from others?” 26 When Peter[k] said, “From others,” Jesus said to him, “Then the children are free. 27 However, so that we do not give offense to them, go to the sea and cast a hook; take the first fish that comes up; and when you open its mouth, you will find a coin;[l] take that and give it to them for you and me.”