That Saint Peter, before he went to Rome, founded the see of Antioch is attested by many Saints of the earliest times, including Saint Ignatius of Antioch and Saint Clement, Pope. It was just that the Prince of the Apostles should take under his particular care and surveillance this city, which was then the capital of the East, and where the faith so early took such deep roots as to give birth there to the name of Christians. There his voice could be heard by representatives of the three largest nations of antiquity — the Hebrews, the Greeks and the Latins. Saint Chrysostom says that Saint Peter was there for a long period; Saint Gregory the Great, that he was seven years Bishop of Antioch. He did not reside there at all times, but governed its apostolic activity with the wisdom his mandate assured.
"That Saint Peter, before he went to Rome, founded the see of Antioch is attested by many Saints of the earliest times, including Saint Ignatius of Antioch and Saint Clement, Pope."
Luke writes in Acts that the church at Antioch was established by witnesses from the Jerusalem church that were scattered because of the persecution after Stephen was martyred.
Act 11:19 "Now they which were scattered abroad upon the persecution that arose about Stephen travelled as far as Phenice, and Cyprus, and Antioch, preaching the word to none but unto the Jews only. And some of them were men of Cyprus and Cyrene, which, when they were come to Antioch, spake unto the Grecians, preaching the Lord Jesus. And the hand of the Lord was with them: and a great number believed, and turned unto the Lord."
No where is it mentioned that Peter ever visited Antioch. When Paul and Barnabas were sent out as missionaries from the church at Antioch, Peter is not even mentioned nor is he mentioned when Paul reports back to the Antioch church.
Act 13:1 "Now there were in the church that was at Antioch certain prophets and teachers; as Barnabas, and Simeon that was called Niger, and Lucius of Cyrene, and Manaen, which had been brought up with Herod the tetrarch, and Saul. As they ministered to the Lord, and fasted, the Holy Ghost said, Separate me Barnabas and Saul for the work whereunto I have called them. And when they had fasted and prayed, and laid [their] hands on them, they sent [them] away. So they, being sent forth by the Holy Ghost, departed unto Seleucia; and from thence they sailed to Cyprus."
Looks like the Antioch church was a self governing body, able to send out missionaries and even voluntarily assist the church at Jerusalem.