This is a (relatively) brief outline of Church-State relations, mostly just the big phases and conflicts, focusing on the West and on the Papacy, from my own perspective, based on my own reading, and for my own purposes:
The birth of Christianity coincides almost perfectly with the divinization of the Roman Emperor. By means of the new Imperial cult, the Emperor was treated as divine or quasi-divine, and the cult of his sacred person and authority quickly became one of the basic cores of Roman and Imperial identity. The Roman Empire, as embodied by the quasi-divine Emperor, was, by this understanding, absolutely sovereign, and not capable of being challenged from the standpoint of divinity, since it was itself, in a very real sense, divine--it also had, naturally, absolute power over religious matters, funding cults and temples and regulating them for its own purposes. Even prior to the Empire, of course, civic and religious life were generally indistinguishable, with political and religious offices and authority going together in most cases.
Christians in the first centuries, though, had a complex relationship with this Imperial ideology. On the one hand, they consistently refused to pay the Emperor divine (or even pseudo-divine) honors, which was one of the primary reasons why they were persecuted. On the other hand, Christians labored to present themselves as good citizens, loyal to the Empire and especially to the Emperor himself--and they sometimes even appealed to the Emperor for internal dispute resolutions, or for aid against local persecution (most persecutions of Christians were local rather than Imperial). As the Church expanded, though, it took on more and more the status of a "society within a society," even an "Empire within an Empire"--the Church as a highly organized institution, shadowing the Roman Empire in all its major cities, participating in its intellectual life and utilizing its infrastructure, but with its own authorities totally separate from, and frequently opposed to, the general public authorities and ideologies. A bishop was a public figure, to be sure, but he was not a civic one--and he represented, in his person, a set of ideas radically different from those animating the state at large. He and the Imperial governor were not likely to get along.