I just spent a week figuring this out after quitting on Civ 3 a number of times over the years, and I'm not that good of a player (obviously), so I'm perfectly qualified for this. All of this advice can probably be found on Civfanatics somewhere, but here it is:
NUMBER ONE THING RIGHT HERE!: 1. There is a quick build option if you go through the setup screen to start a game. Much faster games, like old Civ (not sure if this is just in the patched game).
2. Turn off "show automatic moves" in preferences, this will help in the modern era when you have about 1,000 workers zipping all over giving you a headache.
3. Civ 1 you built every building, every unit, defense was huge. Don't bother in Civ 3. There's virtually no rebellion when your cities are small, so you don't need to build temples in every city. Granary is used for building city size, not for famines (maybe for both?) so it's the most important early building (you'll need lots of workers and settlers). Offensive units are much more plentiful at the beginning of the game, and are far more important than stacking defensive units (because of the borders you can see threats coming and kill them first). Only build 1 or 2 barracks, and use those like unit factories as much as possible. Use a couple cities to crank out money. Use a couple cities to crank out settlers/workers. Specialize a lot.
Overall, in Civ 3, you _must_ be choosy. Don't develop every tech, don't build every building. It's too competitive a game, even on the second difficulty option.
Here's why:
4. Trading is huge-uh! Trade for technologies, you won't get ahead otherwise. Every civilization is competitive in technology and military, so you can't just sit around and build peacefully for 1500 years and expect to keep up like in Civ I. Why? Because the other civs are sharing and aggressive! And they are going to gang up on YOU! So you have to be incredibly active in diplomacy.
5. The AI and the Civilization's borders concept leads to aggressive games. You must focus on building cities for the first few hundreds of years until you've expanded as far as you can on your landmass - every computer player will do this, and then the biggest civilizations will have a huge advantage. This is because of all the diplomacy options related to trading and military (more people=more gold=more military). If you don't participate in trading, it's a huge handicap. The AI all cooperates. Being big makes it easier to play, like having the huge stack in Texas hold 'em. Also, trading keeps them on your side, less likely to gang up on you.
6. The main way in Civ 1 to get tech fast was to explore. That's still in Civ 3, but due to fast expansion, you'll only get a few techs this way. You'll need to get used to hoarding money and paying for techs.
7. Workers and settlers in the first several hundred years. Build mines first (I never delved into the math of the game, I found this advice tremendously useful in the Civ 3 forum).
8. You can't rush Wonders, but you can help them along by clearing forest in the same city and disbanding previously built units. Or if you're lucky enough to get a Great Leader in battle (a totally undeveloped facet of this game). Not a real recommendation, but kind of a trick when you're desperate to finish fast.
9.And choose an "industrious" starting civ, at least until you get used to the new gameplay. They have some advantages that aren't obvious to me (again, I'm completely ignorant of the Civ 3 math, even though I'm a programmer, historian and know statistics, I'd rather just play).
10. I find due to the competitiveness, that I hardly ever have Wonders to work on, and when I do, they are often built first by the other civs. This almost never happened in Civ 1. They're so much more aggressive. Remember, the lesser wonders can be built by multiple Civs. Also in Wonders, there's much more of a bias towards continents - most of the bonuses are continent based. So keep that in mind as you expand, or choose who to invade.
11. Have a tech plan based on Wonders and their bonuses, so you aren't building temples and the Oracle, etc. I loved building everything for high scores, but again, the AI is too good.
Well, the basic advice is to check the quick build option in the setup screen, turn off "Show automatic moves" in the preferences screen (too many workers in Civ 3), expand much, much faster than you're used to, and talk to the computer players as much as possible (you'll even have to capitulate to demands occasionally, which was never the case in Civ 1). Save money for diplomacy and war. Micromanage Workers and the Tech Tree.
