The answer depends, in part, on what difficulty level you're using. Some general advice:
1. Trade for technology. You can't hope to out-research the AI except perhaps at the easiest difficulty settings.
2. Make lots of military units, even cheap ones. The AI seems to assess your military strength primarily based on how many units you have, not how powerful they are.
3. At lower difficulty levels, give peace a chance. Swallow your pride and pay tribute; purchase tech partly with per-turn gold, so that the AI has incentive to stay at peace for 20 years; trade luxuries; etc.
4. At higher difficulty levels, give war a chance. This means securing a source of iron and/or horses to support swordsmen and horsemen. Don't try to build too many wonders; you'll lose the wonder race in most cases.
5. Cash is more important in this game than in prior civs. It's what supports your military units, for example (unlike food, as in prior versions of Civ). So make lots of cash. Favor cash over research; use money to buy tech.
6. When you do have to fight, make sure you have a mix of offensive and defensive units. It's easy to fall into a rut of making only spearmen or pikemen -- defensive units -- but those guys aren't easy to take on the offensive.
7. Hang in there! Civ3 is tougher than its predecessors, but it can be beat, even at the highest difficulty levels. Have fun with it. If the game were easy to beat, it would be no fun.