While Peter is mentioned a lot in the gospels and in the earliest chapters of Acts, often this is because he's the most outspoken and rash of the disciples (Matthew 16:16, Matthew 16:22, Matthew 18:21, Matthew 26:33, Mark 9:5, John 18:10). This is why Peter received so much attention from Jesus (Matthew 16:23, Luke 22:31-34, John 18:11, John 21:15-17). From the second half of Acts forward, however, Paul is mentioned much more than Peter. Paul ends up writing much more of the New Testament than Peter does, and the earliest church fathers (Ignatius, Polycarp, etc.) speak more of Paul than they do of Peter, and they make statements about Paul that are more exalted than what they say about Peter. Paul, by far, receives the most attention early on, even though Peter became more popular among many of the church fathers who wrote from the third century onward.
Did the apostles have any concept of Peter being their ruler? No (Luke 9:46, Luke 22:24, 2 Corinthians 12:11).
Did Jesus think that Peter was a "shepherd" in the sense that he would oversee the other apostles? No. To the contrary, He tells Peter that John's future is none of his (Peter's) concern (John 21:21-22).
The apostles are repeatedly portrayed as being at the same level of authority (Matthew 19:28, Ephesians 2:20, Revelation 21:14). During the doctrinal dispute in Acts 15, Peter's testimony is heard (Acts 15:7-11), but doesn't settle the dispute. James has the last word (Acts 15:13-21), and his terminology is incorporated into the letter that's sent out (Acts 15:23-29). The letter mentions "the apostles and the brethren who are elders", but says nothing of papal authority.
Did the apostles view the Roman church as some sort of "Mother Church" that had supreme authority? No. To the contrary, Paul writes a letter of doctrinal and moral instruction to the Roman church. In his letter to the Romans and in his letters written from prison in Rome, Paul never mentions a papacy, nor does he even mention Peter in association with the Roman church. Paul refers to himself instructing and caring for all of the churches (1 Corinthians 7:17, 2 Corinthians 11:28), something he surely couldn't have done if he didn't have authority over the Roman church. Paul writes about church government over and over again (1 Corinthians 12:28, Ephesians 4:11-12, etc.), but never mentions a papacy. To the contrary, he refers to "apostles" as the highest authority (1 Corinthians 12:28), with no mention of a Pope who is above the authority of an apostle.
Peter himself also had no concept of a papacy. He refers to his authority as an apostle (1 Peter 1:1, 2 Peter 1:1) and an eyewitness to Christ's earthly ministry (1 Peter 5:1, 2 Peter 1:16), but never as a Pope. Although he had just as much apostolic authority as the other apostles, Peter referred to his governmental authority as nothing more than that of a "fellow elder" (1 Peter 5:1). When Peter was nearing death, he said that he was leaving behind written documents in order for people to be able to remember, after he died, what he had taught (2 Peter 1:13-15, 3:1-2). He doesn't say anything about leaving behind a successor, much less a Roman bishop with papal authority.
Obviously, there was no papacy during the time of the apostles, contrary to the claims of the Roman Catholic Church. And it isn't a matter of a papacy not being mentioned just because there was never any occasion for it to be mentioned. If there was a papacy during the time of the apostles, there would have been many contexts in which mentioning it would have been appropriate (Luke 22:24, John 21:22, 1 Timothy, 2 Timothy, Titus, 2 Peter 1:13-15, 3:1-2, etc.). Yet, a papacy is never mentioned. It's even contradicted by Paul's repeated references to his equality with and his independence from the other apostles, for example (Galatians 1-2, etc.). Even if the doctrine of the papacy wasn't contradicted by the New Testament, its absence would be enough to make the claims of the Catholic Church untenable.
(Source(s) Unknown)