With Jerusalem basically out of the picture, it left the Churches in areas like Rome, Antioch in Syria, and Alexandria in Egypt, which coincidentally also happened to be the main political and economic centers of the Empire. (Christianity, after all, was an urban phenomenon.) In fact, each of the three largest cities in the Empire was almost twice the size of any European city at the beginning of the 17th century, and they had a very huge population to boot. (Rome is at the head, boasting about 1,000,000 inhabitants at its peak, followed by Alexandria - 500,000 to over 1,000,000 - and then Antioch - 400,000-600,000.) Other areas, like Ephesus (population: 100,000-150,000), the “largest emporium in Asia” according to Strabo (Geography 14.1.24), Corinth (population: around 100,000), or the reestablished Carthage (population: around 500,000 - second largest city in the western half of the Empire after Rome), also had some level of importance.
Rome, the center of the Empire, of course enjoyed a great deal of prominence (only Alexandria and Antioch could rival it in the affairs of the Christian church in the first two centuries), but it is about Roman Christianity that we have the first and best records. The tradition that both Sts. Peter and Paul - two important figures in the Church, obviously - died there lent spiritual legitimacy to the shift in focus from Jerusalem to Rome. That shift also underscored the fact that Christians began to recognize that the faith’s future lay in the Roman Empire as a whole rather than in Palestine.
The Church in Antioch was at first more important than Alexandria (it was after all, the earliest community outside Palestine and the place where Christians first became known as “Christians”), but by the 2nd-3rd centuries the focus of theological brilliance and creativity began to shift towards the latter. There was a long-standing rivalry between the two sees in the area of doctrinal debate: Alexandria became known for philosophical speculation and the allegorical intepretation of Scripture, while Antioch is famed for theologial conservatism and scriptural literalism. Ironically enough, however, the orthodoxy of two of Antioch’s early scholars (Lucian of Antioch and Paul of Samosata) was questioned, and more variations on the basic Christian theme would come from Antioch than from any other ancient community. (Adoptionism and Nestorianism seem to have had their origins from the school of Antioch.)
Antioch and Alexandria were both important centers of missionary activity: from Antioch, missionaries traveled northeast to Edessa, Asia Minor, Scythia, Mesopotamia and Armenia. From Alexandria, missionaries went southwards, eventually reaching Ethiopia. AFAIK Rome at this particular time period was not so active in that area compared to the two, but then again, opportunities were limited due to on-off persecution. (Note: the widespread, really nasty persecution only began to occur during the late 3rd to the early 4th century. In earlier times, persecution were mainly sporadic, local affairs which usually did not target every single Christian. Heck, Christians probably faced more danger from mob violence than any official action.)
