It was Peter's brother Andrew who first found The Lord after hearing John The Baptist near the Jordan River proclaiming Jesus Christ as "Lamb of God" (John 1:29-36). Andrew then went and brought Simon to Jesus (John 1:41).
It was Jesus Who declared that Simon would be called Cephas, an Aramaic name equivalent to the Greek Petros, from which Peter is derived. Simon gradually became known as Peter among the other people, although The Lord continued to call him Simon when talking to him (Matthew 17:25, Mark 14:37, Luke 22:31).
There has been much controversy regarding the intended purpose of renaming Simon as Cephas, and Jesus' statement that, "And I tell you, you are Peter [Cephas], and on this Rock I will build My church, and the powers of death shall not prevail against it." (Matthew 16:18 RSV). It's helpful to note that Cephas actually means "a small stone," while Jesus Christ is the true Rock Of Ages. The Rock that The Lord was speaking of could only have been Himself, not any mere human. After all, it's called the Christian church, isn't it?
Peter, while often bold and courageous in behavior, never exalted himself. After Jesus' miracle of the large fish catch (Luke 5:1-7), Peter said to Him, "Depart from me, for I am a sinful man, O Lord." (Luke 5:8 RSV). Later, after Jesus' Resurrection when the church was becoming popularly established, "When Peter entered, Cornelius met him and fell down at his feet and worshiped him. But Peter lifted him up, saying, "Stand up; I too am a man." (Acts 10:25-26 RSV).
While Peter was most certainly a prominent and important member of the new Christian church, at no time did he exercise, or claim to have, authority over the other apostles (see the Fact Finder below). Many people are quite surprised when reading European history that the "Primacy of Peter" doctrine was established centuries after Peter died as a tactic in a political struggle for power between numerous bishops and church authorities of that time. It did not originate anywhere in The Holy Bible.
A summary of Peter's activities during Jesus' ministry:
Fact Finder: Did Jesus Christ elevate Peter over the other apostles?
See Did Peter Have The Primacy?