A few suggestions if you don't mind. Please note, I'm no expert on culture victories, so take this as you wish.
(and then get overrun by barbarians when I only have 3 warriors guarding two cities in 1250 BC)
First off, why do you have barbs turned on? Since I'm trying to get my three cities up fast I don't want the worry of barbs, so they're turned off. Remember, according to the
HoF Rules, barbs are optional.
The religious route where I try to found Polytheism and then build an early Stonehenge and Oracle.
I don't bother even trying for the early religions. IMO it's a waste of turns when I could be getting Pottery, Writing, and Alphabet. Once I have Writing I sign OB with anyone who has a religion in the hopes that it'll pass my way. Granted, some times I have to wait a long time for it to happen, but its still not worth researching it IMO. Also, I stay clear of most wonders as I don't want to pollute my

pool with non-artist points. The only wonder I build that doesn't provide artist points is the Pyramids for the government civics. Other than that, other wonders aren't worth it IMO.
The non-religious route where I focus on Bronze Working and land improvement techs to get capital up and running
Remember your main focus on this game is culture, not military. IMO there is no reason to bother with the metal techs, except for forest chopping. I've found that the AI tend to trade BW, so I see no reason to research it myself when I can get it in trade later. IW is the same way. I may have to wait longer to get it, but that doesn't seem to be a problem. Honestly, I'm always way behind in military strength compared to the AI (I'm typically in last place), but it doesn't cause a problem. Between choosing the peaceful opponents and my trade bonuses I'm usually well liked.
The religious route sounds like a good idea in a culture game but somehow it never seems to generate very good results.
I've learned this myself, hence my reason now that I don't even bother.
Hopefully the expert culture players come along and either give better advice, or point out my errors. Granted, this early in the competition, they might be hesitant.
