There is really no excuse for letting your own cities get converted. The computer seemingly stops after the Renaissance.
If you can survive the torrent of conversion attempts into the Industrial Age when you HAVE to spend faith to get Prophets, you can pretty much push your religion everywhere. I converted a couple of whole kingdoms before they had a chance to react. Great prophets are crazy powerful against tall civs that aren't paying attention. All of the other cities, including the holy cities, had fallen under the unceasing pressure of my religion. Last game I finished there was only one city not following my religion, and that was a holy city (and even it was teetering). The AI is terrible at religion management too.
That gold adds up quick once you start converting the world. I like to go for Tithe as a religion bonus. Even small maps on quick speed take a long time, at least by my standards. It was fun watching every other civ immediately denounce me after I'd won the World Leader vote but before the turn ended and it went into effect. Second time the World Leader vote came around I had the delegates to take it. I'd been teching toward a science victory (and in fact, that was the level-3 Freedom tenant I'd picked first), but I about-faced and went toward Globalization (which would've put me well over) only to realize that apparently just coming close to winning a diplo victory gets you a few extra votes. Oops.) I still only had 34 of 35 needed votes for the diplomatic win when it hit United Nations, even as host, with all City-States allied, and both World Religion and World Ideology.
After I had pretty much full control, I rammed through my World Religion (which was about two thirds of the world by that point, so most other civs were okay with it) and then my World Ideology (Freedom, which no one else at the time was - England declared themselves for it soon after, but only to be immediately conquered by Denmark. I was rolling in cash from trade routes, so I ended up bribing a bunch of City-States into my corner as a purely defensive measure (Denmark was snatching them up and would have been able to ram whatever he wanted through the World Congress), but ended up snagging them all and using that for a diplomatic victory. Messed with tourism again and actually understood the Great Works mechanics for the first time, which was nice, but by that time Denmark was already kicking my ass in culture and I was too far behind to catch up. That one wasn't quite as smooth because I sort of waffled on what victory condition I wanted. Second game I was the Ottomans on an Archipelago map. ) After that I just focused on tourism and not pissing anyone off badly enough to attack me and eventually got my culture win. (Just in time, too - my new borders pushed his settler back away from the spot. I had a whole corner of a continent to myself my only neighbor was Japan, who I managed to block by dropping a city on an isthmus and refusing to let him past.
Kept things tall, rather than wide (which is my usual preference) after seeing some discussion of it here it worked out pretty well for me. Played around with BNW's new tourism system and ended up winning a culture victory. First game I ran as the Iroquois, ended up on a Continents map.
Random map, small, Prince difficulty (yeah, I know, I'm a wuss ), quick speed, all victory types on except time. So I got a chance to play a couple games this weekend.