Peter never did build the church or a church. Jesus is the one who built the church.
Answer
The Apostles founded the early Christian churches. The missionary journies by Paul and Peter (and other apostles which were not documented) helped establish and grow the churches. This all began within the years after Jesus was crucified or after 30-40 AD. Some of Paul's later writings after this time was an encouragment and admonishment of the Churches.
Peter was thought to have started the "first church" because of the interpretation of the scripture in the New Testament where Jesus spoke to Peter "Upon this Rock I will build my Church". This is a view generally accepted by the Roman Catholic Church. However most scholars do not interpreted this scripture to mean only Peter founded the Christian Church.
Mt. Zion
St. Peter the Apostle Church was created in 1842.
Peter was the apostle chosen by Christ to be the head of his church. Jesus referred to him as the "rock" upon which he would build his church, and he gave Peter the keys to the kingdom of heaven. Peter played a prominent role in the early Christian community and is considered the first Pope by the Catholic Church.
Peter the Apostle, also known as Simon Peter of Simon Barjona was the rock upon which Christ built his Church.
According to the Nag Hammadi, it was Mary Magdalene because she was the first to have been approached by the risen Christ. The Nag Hammadi library are Gnostic texts from around 200 AD. Some historians believe they predate some of the main texts of the Bible.
Peter I think
Saint Peter the Apostle
St. Peter the Apostle and first pope of the Catholic Church.
The Papal Basilica of Saint Peter
The patrons of the Universal Church are St. Joseph and St. Peter the Apostle.
Christ chose the Apostle Simon, who he renamed Peter, as the first leader of His church.
According to historical reference, St. Apostle Peter was the first leader or pope of the early church. As the pope of the church, he was the one regarded as its first office-holder up to the recent times.