Antioch, the capital of Syria in Seleucid and Roman times (Damascus was an older city, but was somewhat smaller than Antioch for a while), is where the first sizable contingent of non-Jews converted to a belief in Jesus, and where according to Acts the name "Christian" was first used. An early council in Jerusalem (reported in the "We" document, the most reliable part of Acts) decided that the Gentile Christians were not subject to the full Jewish Law (in particular, circumcision was not required, which might have limited the appeal of Christianity to males). Antioch was the launching point for missionary ventures like Paul's, and since seniority was the basis for leadership among the churches, the church of Antioch had a good claim to govern all of Gentile Christianity.
Indeed, early patriarchs of Antioch claimed that they, not the bishops of Rome (the term "Pope" was not widely used until the 4th century), were the successors to Peter. This claim to "succession" was somewhat weakened because the first two were not on record as saying such a thing. When Peter went to Rome (probably not expecting to die there!) he left a certain Evodius in charge. But we know almost nothing about him except the name, not even whether he was Gentile or Jewish (one tradition has him being the first pagan to convert, which doesn't fit well with the story of Peter and the centurion; but another has him as one of the "seventy" appointed by Jesus as missionaries in addition to the 12; neither of these is at all trustworthy); or whether that name was a given name or a nickname: if a nickname, since it means "good song" perhaps his talents were more musical than literary, which would explain why we have no writings from him. He lasted for decades, unlike many of the early Christian leaders, and there is no martyrdom story although he appears to have died around the time of Domitian's persecution. His successor Ignatius, however, wrote a large number of influential epistles around the turn of the century.
Early Christians thought the epistles of Ignatius were so good that there ought to be more of them. Pseudepigraphy (writing new texts in the name of old authors) is a common problem in early Christian literature. We have the "short recension" of Ignatius (excerpts from numerous letters all put together in Greek), the "middle recension" (letters separated out and given in full, also in Greek), and the "long recension" (in Latin, with more letters, and extra material inserted into the other letters). The long recension is certainly full of forgeries: it includes Latin words for institutions and theological concepts that did not exist in Ignatius' time, so the new material cannot be a translation from the Greek. Some scholars think all of the middle recension is genuine; some that we can only trust what is in the short recension; some that we cannot trust any of it. I think the middle recension is at least majority genuine. It is noteworthy that Ignatius does not present himself as any kind of authority: he urges unity among the "catholic" church against the dubious gnostic sects (katholikos first appears as the designation of the mainstream Christian church in Ignatius), and tells the priests (or diakonoi rather: they are still called by this word for "waiters", as the servants who distribute the bread and wine, rather than by any term like "priest" suggesting a particular holiness) to obey the bishops (episkopoi "overseers") in matters of doctrine and discipline. Yet he does not rank himself among the bishops, using the phrase "my fellow servant" whenever addressing another diakonos, as if he himself belongs to the lower rung, rather than the higher, of this nascent hierarchy.
However humble Ignatius may have been, subsequent leaders in Antioch did claim the top spot in the church. Theophilus (patriarch in the mid-2nd century) is addressed as "Your Excellency" in the cover letters which open the "gospel of Luke" and "book of Acts" as we now have them. Theophilus was forced to respond to Marcion, a teacher whose scholarship was acknowledged by everyone but whose views were quite radical. Marcion published the first "canon" of Christian texts: the Evangelion, Apostolikon, and Antitheses. The Evangelion was like "Luke" except that it was just called "the gospel" without ascription to Luke or any other particular author (Marcion did not claim to have written it himself, and probably didn't, since much of it is not very compatible with his views), and lacked the opening chapters (the nativities of John the Baptist and Jesus, the baptism, and the temptation in the desert: it started "The Son of Man came to Capernaum" with Jesus reading Isaiah in the synagogue and starting his ministry by saying "This day these prophecies are fulfilled"). Orthodox writers naturally claim that the full "Luke" must have existed first, and that Marcion must have suppressed those chapters, but this will not wash: there is simply no reference to anything like the third gospel existing before Marcion's publication, and if Marcion were going to delete anything from the book, those chapters are not what he would have objected to.
The Apostolikon was a collection of ten Pauline epistles. Five had been circulating in Greece and are quoted in other early Christian authors prior to Marcion: 1st Thessalonians, 1st and 2nd Corinthians, Phillipians, and Romans (not addressed to "Romans" in Marcion's text, which again did not include all the chapters we have now); Marcion also included 2nd Thessalonians, which is certainly a forgery that he was taken in by (it presumes that collections of Paul's epistles are already circulating, and that people are already forging letters in Paul's name, neither of which is plausible in Paul's lifetime). Marcion scoured Asia Minor to find some others, which had not been previously well-known: Philemon (certainly genuine), Galatians (only some minor tamperings, in a couple of verses referring to Petros "Peter" where the real Paul always says Kephas instead), Ephesians (much debated, but probably contains extracts of a couple different Pauline letters plus a lot of non-Pauline material; Marcion's text was addressed to the "Laodiceans" instead and had some differences), and Colossians (probably not by Paul at all). Marcion did not have 1st or 2nd Timothy, Titus, or Hebrews: none of those were by Paul, although Hebrews is by somebody in the 1st century.
The Antitheses presents Marcion's theology. God's first creation was the Demiurge, the agent through whom all the other work of creation was accomplished; this was not new: the Logos "Word" in Philo of Alexandria and the gospel of John has this role. Marcion however believed that the Demiurge turned evil, through the sin of pride, unable to perceive that there was anything higher than himself and therefore acting against God. This became a very influential concept among many Gnostic sects, but not all of them took Marcion's next step: he identified YHWH in the Old Testament as the evil Demiurge, although thinking that other parts of the Old Testament were referring to the true Most High God; the role of Jesus was to free us from the tyranny of YHWH. The "antitheses" to which the title referred were a selection of "contradictions" between the loving teachings of Jesus and bloodthirsty passages in the Old Testament (of the sort that you can find cited in "angry atheist" literature nowadays). Theophilus fought against this anti-Semitic version of Christianity, but wanted to salvage the valuable texts of the gospel and epistles which Marcion had put together: it is from his efforts that the orthodox "canon" began to take shape.
Antioch's leadership role began to decline subsequently, however. A low point was Paul of Samosata, a mid-3rd-century patriarch who promoted an "adoptionist" theology in which Jesus had just been an ordinary human until God raised him to divine status at the time of the baptism. The original text of "gospel of Luke" (as published under Theophilus; Marcion's Evangelion did not have the baptism sequence) correctly quoted Psalms 2:7 "This is My beloved son; this day I have begotten thee" (as also quoted once in Acts, referring to every believer, and twice in Hebrews, referring to Jesus) but many manuscripts changed it to "This is My beloved son, in whom I am well pleased" to avoid precisely the implication which Paul of Samosata insisted on. He also thought the Holy Spirit had once been an ordinary angel until God raised him to divine status: he was not teaching Trinitarianism but rather Tritheism, that these three were fundamentally different Gods, only one of them existing from all eternity. Syrian bishops got together, deposed him, and elected Dominus in his place. This was during the chaotic "barracks emperor" period of frequent military coups in the Empire, and queen Zenobia of Palmyra (an Arabian kingdom, tributary to Rome) was an independent power in the East (a formidable woman, proudly descended from a daughter of Antony and Cleopatra, she had taken territory from Sassanian Persia, purportedly on behalf of Rome). Paul appealed to her, and she invaded Syria and took Antioch on this pretext.
Zenobia found so little resistance from the disorganized Roman forces in the East that she moved on to take Egypt, where she was hailed as a native daughter come home (the Cleopatra connection), and when forces from Asia Minor tried to re-invade Syria, she ended up with a good chunk of Asia Minor too. But a Western emperor named Aurelian was re-unifying the realm, putting down a separatist line of emperors in Gaul (the city of "Orleans" was his foundation) and turning finally to Zenobia. Her son he had to kill, but she was retired honorably to a villa, with a new Senator husband, where she conducted seminars on her own unique version of Christianity. Paul of Samosata then sued in the imperial court to get the patriarchal palace in Antioch back! Christianity was still only semi-legal (there was a "toleration" edict in place), but Aurelian was trying to reconcile everyone and was not at all a persecuting sort. He ruled that it was the property of the Christian organization as a whole, and that the bishop of Rome was the proper spokesman for that organization. Felix, the Pope (as I suppose we can begin calling the bishop of Rome, with this legal endorsement of supremacy), ruled in favor of Dominus.
The patriarchate of Antioch shattered completely during the theological controversies over the "Council of Chalcedon" which defined Jesus as one Person who had two Natures. The Nestorian view was that there were two different Persons, the human carpenter's boy and the divine Word of God for whom the human acted as a willing channel (in particular, Nestorius thought that any special veneration of his mother Mary was improper; the title Theotokos "bearer of God" which was popular in Alexandria was a total blasphemy, Nestorius said); this was popular in Asia Minor. The Monophysite view was that Jesus had only one Nature, totally divine; this was popular in Egypt. In Syria, both forms of opposition had a following, as well as the orthodox Chalcedonian compromise. The Nestorians largely fled across the eastern border to Persia, where their patriarch had his seat in Nisibis; they grew into an enormously large church in Asia, with converts as far east as Mongolia, but were almost entirely wiped out in the time of Tamerlane, and are now the "Assyrian Church" in Iraq, the smaller of the two Christian denominations there, with a couple thousand surviving members.
The Jacobites (after an early patriarch of theirs), who leaned toward the Monophysite direction, also outnumbered the Melkites "king's men" who obeyed the imperially-sanctioned patriarch. Some of the Jacobites also went east, becoming the larger "Chaldean" denomination in Iraq, but others maintained a semi-underground presence in Lebanon and Syria until the Muslim conquest, when they didn't have to hide anymore, and became the dominant "Maronite" church in Lebanon (there are also some "Greek Orthodox", that is "Melkite", Christians in Lebanon). Both the Chaldeans and the Maronites are now in communion with Rome, accepting the supreme leadership of the Pope. In early medieval times, the Maronites were responsible for a theological innovation called the "Monothelite" position, which some thought a good solution to the puzzle about Jesus, and others thought a dangerous heresy: it means "one will", teaching that although of course there was a difference between the humanity and divinity in Jesus, Jesus had completely surrendered his human will to the divine will, and in that sense was unified. The 9th-century Agatho accepted this position, but his successors agreed with a council condemning the Monothelite doctrine, and had to sign a humiliating acknowledgement that Agatho was a heretic (the only Pope condemned by later Popes as a heretic). Nonetheless, eventually Rome and Lebanon reached some kind of weasel-wording about the senses in which the "one will" could be considered correct.