I play on Monarch, because the tool-tip said that was where the AI was just plain smarter. I got stomped about 20 times in a row, mostly as the Romans. Now I'm finally into the Industrial era (as Catherine of Russia) and only in 3rd place.
Things I learned:
1. Creative is uber. In the beginning you are always, always trying to grab land before they do. This early-game boost is just overwhelming, in my opnion. "Land is the foundation of the State!"
2. Financial is awesome. Organized is pointless; why save money on expanding to more cities when you could just make more money in the first place? You can start doing this with your first cottage (if it is on a river).
3. Founding a religion is really, really great. I always go stright for Polytheism, and if I miss it, that's ok - I go to Monotheism (the Judeaism audio track is cooler, anyway). Convert to your new-found religion immediately - its +5 culture in the founding city, which is huge in the beginning. I spend most of the game in Organized Religion civic.
4. Pottery is the uber-tech. You are going to make cottages. Lots of them, and early. After I get my religion the first thing I go for is pottery.
5. Archery is like, necessary. If you don't have lots of archers the AI will push you around. And barbarians will eat you.
6. Copper is better than Iron, since it comes earlier; but you will need one of the two as soon as possible. You have to have spearmen to stop their rampaging chariots and axemen to stop their swordsmen.
7. The pillaging! The point of going to war with your neighbor is not to capture his cities - that's incredibly hard - but to burn his infrastructure to the ground. Those towns take 70 turns to mature, and you can burn them in 2 turns with a horse unit.
8. Aztecs suck. If you start next to the Aztec, they will attack you early and often, no matter how pointless. They can't win, but they can gaurantee you don't win.
9. Start making Great People early, preferablly in one city. The scientist Academy is incredible, and founding your religion's special temple helps it spread faster.
10. I always get my religion, archery, pottery, and then I go for the Alphabet. I usually get it first, and then I can trade around and catch up with the AIs.
11. Always, always, always have your scout end his turn in a forest or a jungle. It's his only hope of living through the night. And of course, just like Sulla showed us, his first two promotions are Woodland I & II, for the movement benefit.
Traits
All the other traits suck, since they don't help you on turn 50. Maybe Industrious or Philosophical; but Expansive and Organized are just lame. +2 Health only helps halfway through the game, and Organized is almost as bad, plus it hardly helps. I think these two traits exist for the Chieftan 100 cities micromanage love fest game. Which is a style I sometimes enjoy.
War
I never build Knights; I have an early warfare period with axes and horse archers, and a late period with Cossacks. Knights would be cool but they just don't fit in. Even my Cossacks have like 10 turns where they rule, and then Riflemen show up. Units become obsolete faster as the modern age approaches.
I stopped building walls; with Creative, my cities get to 40% cultural defense pretty soon, and walls are only 50% (and they don't stack!). Catapults, however, are necessary if you want to take another city.