Peter is the best-attested of the twelve, and the one whose later significance rests most heavily on a single sentence.
All four Gospels place him first among the disciples, give him the name change from Simon to Kephas, Aramaic for rock, rendered into Greek as Petros, and record his denial of Jesus during the trial, three times, before a cock crowed. The denial is in all four, which is a strong indicator: it is not the sort of detail a movement invents about its senior figure. Paul independently confirms his standing, calling him Kephas and describing a public argument with him at Antioch over whether Gentile converts had to keep Jewish food law. That letter is the earliest surviving Christian document mentioning him and it shows him losing an argument, which is again not an invented detail.
The sentence everything else hangs on is Matthew 16:18, where Jesus tells him that on this rock the church will be built and gives him the keys of the kingdom. The Roman Catholic reading takes the rock to be Peter himself and the keys to be a transferable office, inherited by the bishops of Rome. Orthodox and Protestant readings variously take the rock to be Peter's confession of faith rather than his person, or accept a personal primacy of honour without an inheritable jurisdiction. The Greek plays on the pun, Petros, petra, and the pun works better in Aramaic, where both words are kephas, which is itself an argument that the saying is early.
What the New Testament does not say is that Peter went to Rome, led its church, or died there. That belongs to tradition.