From the book of Acts, it is evident that the church began as a Jewish entity. The life of the body is summarized in Acts chapter 2.
Acts 2:41-46 (NASB)
So then, those who had received his word (Peter's) were baptized; and that day there were added about three thousand souls.
They were continually devoting themselves to the apostles' teaching and to fellowship, to the breaking of bread and to prayer.
Everyone kept feeling a sense of awe; and many wonders and signs were taking place through the apostles.
And all those who had believed were together and had all things in common;
and they began selling their property and possessions and were sharing them with all, as anyone might have need.
Day by day continuing with one mind in the temple, and breaking bread from house to house, they were taking their meals together with gladness and sincerity of heart,
Of special notice in verse 46 is the mention of going to the temple every day. The earliest believers did not consider themselves to be a part of anything new. They were just another sect of Judaism.
With the martrydom of Stephen and the ensuing persecution, the Jewish church at Jerusalem experienced it's first scattering. Positively, Philip took the gospel to Samaria, while negatively, Saul went to Damascus, a city in Syria, with letters enabling him to arrest and capture any believers in Yeshua that he might find there. Peter then received the vision from God that salvation was also for the Gentiles.
Acts 11:19-20 (KJV)
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.
Antioch became the first place where the gospel was purposefully preached to Gentiles. It also became the base for the missionary journeys of Paul, formerly known as Saul and persecuter of the church, and his companions. It is at this point in Acts that the emphasis switches from Peter and the Jewish church to Paul and the Gentile church.
Because of this shift in attention, it almost seems like the Jewish church was left behind. The influx of Gentiles into the body of Yeshua became overwhelming to the point that the church eventually seems to have morphed into a very different entity than the Jewish church that started it. However, this did not happened during the book of Acts. When Paul was called into question by some from the Jerusalem church, he turned to James and the Jerusalem Council. When Paul returned after his third missionary journey he submitted to James' suggestion for the completion of his nazarite vow that he also pay for four others to prove that he was not trying to subvert the Old Testament law.
So what did happen to the Jewish church and how did Christianity become so different, so not Jewish? Stay tuned for part 2!