I sympathize. I also had a steep learning curve when I upgraded. The key to doing well in Civ2 was building camels and sending them out. In Civ3 you can't buy wonders and the AIs won't sell you their cities so easily. Trade is a side show in 3; the only thing you as ruler worry about is the trading of big items like techs and luxuries. Luxuries are important to Civ3. Try to keep a good supply to keep your citizens happy. Citizens in cities clinging to their base cultural identity means that, even if you did buy it, there's a chance they'd flip back to the country you bought it from.
In Civ3 keeping your citizens happy is much more important and building wonders is pretty optional. Just build the ones that will help you get specific things done, like lower unit costs or generate extra shields, gold, or techs. Sometimes I get pretty frustrated and just go out conquering the enemy. The culture victory is probably the easiest win strategy. Just build temples & libraries and focus on driving down your corruption so you can build more buildings.
Wars are much easier to manage. When you get one, just ally up with everyone who surrounds your enemy. You won't have to conquer too much before they're begging for peace. It's better to let your allies crap out on you first and make their "separate peace" (or if your citizens can take it, wait out the full 20 turns and non-renew your alliances) so that your reputation doesn't take a hit. It's near impossible to go through a whole game without some wars, but I find that I can avoid fighting in Civ3 easier than in 2, plus the AIs doing all just gang up on me like they did in 2.
I'd say run up the tech tree as fast as you can and don't be afraid to trade away techs to stay on good terms with a neighbor. Build embassies ASAP so you can keep track of what's going on in your neighbors' capitals. Drive corruption down (you can ride Republic all the way to the modern age if you're not too belligerant), but don't be afraid to swap over to monarchy or communism when you need to fight a war. I understand fascism is good for wars, too, but I'm too sentimental to ever try that. When you want a big push for expansion, totally conquering another civ is preferable to letting a small bitter enemy linger to harass you for the last few centuries. But overall, keep your wars short. With enough luxuries and (if necessary) cathedrals, you'll find wars are much easier to manage in a democracy. Set your game off of "always renegotiate trade agreements" at first. AIs will hang onto trades with you even if they're not pleased at the moment, but if you renegotiate terms, you could lose out. But if they're happy & prospering, you want to be able to raise your price or pull down an extra tech when you're ready to renegotiate, not just when you hit that 20-turn magic number.
It's true you have to think differently in Civ3. Just make sure you're playing on regent level or higher. If you practice playing on chieftain, you'll teach yourself the wrong way to play.
I personally had an even bigger problem adjusting to Civ4. I don't know how to make my subjects happy. In fact, a lot of what goes on in the game is still a mystery to me. As it stands, it doesn't give me the same epic feeling I got from 2 & 3. So I'm mostly sticking with Civ3 until I can devote more learning curve time to it in the summer.
Is there a way to speed up wonder production that I'm unaware of?
Is there a way to take over foreign cities without military action? (I'm already aware that sometimes they switch allegiance voluntarily when they are impressed by your culture.)
The best way to nail down the few wonders you want (and you must reduce your personal demand for achieving them) is to build big cities with high shield production and a try to get a jump start on the footrace to your wonder of choice. Hitting techs early so you can generate a Great Scientist is the only way to rush-build a wonder. Make one of your cities a "Super Science City" (with the colossus, Copernicus, Newton, a university, and a LOT of trade). But the only wonder you really HAVE to build is Hoover Dam. Build that, and you'll probably win the game.
I think if you turn off the Great Scientist option, then Great leaders generated by battles can still rush-build wonders, but I may also be wrong about that.
Flipping cities by culture is a slow process, but pretty satisfying. AIs treat selling cities almost as taboo. You'll have to fight a few wars to round out your empire, I'm afraid. Just keep 'em short.
I could be wrong, it's been years since I've played it.
