I play at emperor, Civ 3. I've modified the game, but I don't my changes affect this.
1. As soon as you can research polytheism, go for it. Don't be intimidated by the 28 turns or whatever it'll take. You'll beat the AIs every time, and can trade it for 2 advances and gold, or an advance and a contact. Do that 5 or 6 times, and suddenly you're in great shape.
2. Catapults are useful. Artillery are useful. Cannon are not. OK, they do some good against musketmen, but that era passes quickly, and the damn cannon seem to "fail" way too often. Never build cannon; as soon as they're available, disband all of your catapults.
If you sense a war coming, either by your choice or the AIs, set a couple of cities without barracks to building catapults. This is esp. helpful if your plan is to attack an important enemy city in order to get a wonder. The city will be well defended. At least until your 4 catapults unload.

We all know the value of artillery, but IMO catapults are almost as good in the ancient era.
3. My best games come when I can get a settler/worker factory going early. For example, I've been trying to win as the Japanese. I've been close, but just missed. This time I think I'm going to win, and I think the biggest reason is that Kyoto had 3 grassland cows, and is on a river. I irrigated the first, and mined the next two cows. I got lucky early in the game in that I was founding my 6th and 7th cities with just a warrior for defense. I was expanding so fast my military couldn't keep up. But once I had my "territory" I caught up my military and was in great shape.
I even took Kyoto out of "expansion" mode and built 2 wonders there.
Anyway, this isn't the first time I've had my first successful game with a civ because I got an early settler factory going.