There is ample evidence in the New Testament that Peter was first in authority among the apostles. Whenever they were named, Peter headed the list (and Judas Iscariot was always named last):
1 Then he summoned his twelve disciples and gave them authority over unclean spirits to drive them out and to cure every disease and every illness. 2 The names of the twelve apostles are these: first, Simon called Peter, and his brother Andrew; James, the son of Zebedee, and his brother John; 3 Philip and Bartholomew, Thomas and Matthew the tax collector; James, the son of Alphaeus, and Thaddeus; 4 Simon the Cananean, and Judas Iscariot who betrayed him.
16 (he appointed the twelve: ) Simon, whom he named Peter; 17 James, son of Zebedee, and John the brother of James, whom he named Boanerges, that is, sons of thunder; 18 Andrew, Philip, Bartholomew, Matthew, Thomas, James the son of Alphaeus; Thaddeus, Simon the Cananean, 19 and Judas Iscariot who betrayed him.
13 When day came, he called his disciples to himself, and from them he chose Twelve, whom he also named apostles: 14 Simon, whom he named Peter, and his brother Andrew, James, John, Philip, Bartholomew, 15 Matthew, Thomas, James the son of Alphaeus, Simon who was called a Zealot, 16 and Judas the son of James, and Judas Iscariot, who became a traitor.
When they entered the city they went to the upper room where they were staying, Peter and John and James and Andrew, Philip and Thomas, Bartholomew and Matthew, James son of Alphaeus, Simon the Zealot, and Judas son of James.
sometimes the apostles were referred to as “Peter and those who were with him”
Peter and his companions had been overcome by sleep, but becoming fully awake, they saw his glory and the two men standing with him.
Peter was the one who generally spoke for the apostles
Then Peter approaching asked him, “Lord, if my brother sins against me, how often must I forgive him? As many as seven times?”
And he asked them, “But who do you say that I am?” Peter said to him in reply, “You are the Messiah.”
Then Peter said, "Lord, is this parable meant for us or for everyone?"John 6:68-69
68 Simon Peter answered him, “Master, to whom shall we go? You have the words of eternal life. 69 We have come to believe and are convinced that you are the Holy One of God.”
and he figured in many of the most dramatic scenes
Peter said to him in reply, “Lord, if it is you, command me to come to you on the water.” 29 He said, “Come.” Peter got out of the boat and began to walk on the water toward Jesus. 30 But when he saw how (strong) the wind was he became frightened; and, beginning to sink, he cried out, “Lord, save me!” 31 Immediately Jesus stretched out his hand and caught him, and said to him, “O you of little faith, why did you doubt?” 32 After they got into the boat, the wind died down.
24 When they came to Capernaum, the collectors of the temple tax approached Peter and said, “Doesn’t your teacher pay the temple tax?” 25 “Yes,” he said. When he came into the house, before he had time to speak, Jesus asked him, “What is your opinion, Simon? From whom do the kings of the earth take tolls or census tax? From their subjects or from foreigners?” 26 When he said, “From foreigners,” Jesus said to him, “Then the subjects are exempt. 27 But that we may not offend them, go to the sea, drop in a hook, and take the first fish that comes up. Open its mouth and you will find a coin worth twice the temple tax. Give that to them for me and for you.”
23Jesus looked around and said to his disciples, "How hard it is for those who have wealth to enter the kingdom of God!"
24 The disciples were amazed at his words. So Jesus again said to them in reply, “Children, how hard it is to enter the kingdom of God! 25 It is easier for a camel to pass through (the) eye of (a) needle than for one who is rich to enter the kingdom of God.” 26 They were exceedingly astonished and said among themselves, “Then who can be saved?” 27 Jesus looked at them and said, “For human beings it is impossible, but not for God. All things are possible for God.” 28 Peter began to say to him, “We have given up everything and followed you.”