Ok - Played til 3am last night and have a few suggestions.
1. Don't play on too high a difficulty level. Yes, you may "own" Civ4, but play one on easy and get the hang of it. Winning is more fun than losing.
2. You can't win by excelling at one facet of the game. If you don't balance technology, military, economy, City States and happiness, you are going to lose. No more falling behind on technology and trading to make up. You need to consider what to build in each city very carefully.
3. Warfare is your edge. With a combined arms army you can do much better than the AI, but don't get cocky. If you overreach, the AI is good enough to take you out.
4. The game seems to pick a lower resolution than your computer is really capable of. If you have a card with lots of memory, but that isn't fast, you may be able to bump it up.
5. Wonders are much more of a risk. The AI builds them and it is no longer a way of "producing" money before you get the tech.
6. ALWAYS make sure you have enough money! You can ALWAYS buy your way out of any other trouble, but there is no way of curing a money shortage.
7. You are not going to win this game by the middle ages like lower levels of Civ4. This one will take time to unwind. Having said that, NORMAL speed is slow enough. I couldn't stand Civ4 on anything faster than epic and normal is plenty slow enough to strategize.
8. I used to pack Civs into Civ4 to make the game more interesting. With the city states, the regular density of Civs seems fine.
9. Take time to enjoy diplomacy. Just because you can't see the modifiers, doesn't mean you can't negotiate. Research agreements are very profitable. Build friendships with powerful Civs and unite against the Civ you want to take out. Relationships develop more slowly than Civ4. You can't make allies as quickly, or appease the other Civs if you are really in conflict over land. You can't afford to give away resources as much as you could in Civ4.
I'm enjoying it immensely. It suits my style of play, far more than Civ4 did, and I've already deleted Civ4.