The first thing I do with any new 4X is play through once on the easiest difficulty level, just to get a sense of the shape of the tech tree and the basic rhythms of gameplay. It's an approach with upsides and downsides - few games put their best foot forward on easy mode, but higher difficulties can be needlessly frustrating if you're going in blind. In general though, I find that an early win helps me bond with a game.
That being said, Civilization III is more or less exactly as I remember it. The parts of the game I didn't care for, particularly the stingy distribution of resources and the "dead techs" on the tree that you have to research, but don't give you anything interesting, are still there, and there's a couple more annoyances that I either didn't notice or forgot about - parts of the interface require multiple mouse clicks where in later games they'd only need one, and managing pollution is a nightmare because the worker automation cancels whenever they finish. Still, it's a Civilization game, and that's a hard formula to mess up (I think in the future, I'll just have to try and win before the industrial era).
The thing about Civilization III is that it really did introduce a lot of good ideas that future Civ games would develop into incredibly rich mechanics - cultural borders, unique abilities to differentiate civilizations, strategic resources - but with the first iteration of these ideas, they have not yet been fine-tuned, so they come off as kind of clunky. The cultural victory is passive and boring, the unique abilities are all over the place, balance-wise, and strategic resources are a bottleneck that cuts you off from huge chunks of the tech tree and leave you distressingly vulnerable if you get a bad draw. I think I'll play a couple more games and then try and see about downloading a mod. I'm certain that somebody, somewhere must have gone through the trouble of tweaking the gameplay to make Civilization III into the great game I know it has the potential to be.