I hardly qualify as an expert, but I do beat Monarch and Emperor level consistently, though I struggle at the highest levels. People have already made most suggestions I would make, but I would add a couple:
1. Don't save-and-reload. It will get you into bad habits, and if you ever try to compete in the Game of the Month, you'll be at a big disadvantage. Even if you don't want to compete, you'll develop sloppy habits -- making ill-advised attacks because you know you can always reload, timing tech trades in a particular way because you know you can undo your mistakes, etc. It's more satisfying to win after you've taken your share of lumps and fought back. And save/reload is the one cheat that is NOT available to the AI.
In fact, I wish Civ 3 had an "Iron man" setting akin to that of Alpha Centauri, one that precludes you from reloading.
2. To echo Ision's post from another thread, try not to get in the habit of "always" doing something. That approach again will ultimately limit your abilities. Play different civs to try to win with different traits; pursue different types of victories; build different wonders; play as a builder, warmonger, culture-monger, scientist. Lately I've taken to setting everything -- even my own civ -- to random. I'm also going to try the GOTM.
These two suggestions not only will help you improve, but will also make the game more fun, IMHO.