I am actually rather proud of myself, for figuring out how to fix Civilization III to work in Vista. Don't know if this would help a lot of other folks, but it worked for me. I know some folks have no problem running Civ III in Vista, some do, and some have had luck with other fixes.
I mentioned my troubles with Civ III on Twitter, and got what seem to be the most common solutions: compatibility modes, run as Administrator, and so on. None of these helped.
Then I thought about what exactly was happening. The program would start, show the splash screen, then immediately stop. What happens after the splash screen? Two AVIs are shown, one the company logo, one "dramatic" movie about Civilization. So, I'm thinking graphics.
Tried manually reducing the resolution of the screen to the appropriate setting for the game. Small improvement; a black screen with some of the text menu items showing, including, luckily, the "Exit" button. So, I think, what does Civ III use to "talk" to the graphics card? DirectX. It installs, or tries to install, DirectX 8.x as part of the game install. Busted DirectX, maybe?
Uninstalled CivIII. Downloaded and installed the newer DirectX 9c. Reinstalled CivIII. And it works. No run-as-Admin, no need for compatibility mode (though I might try that again to improve smoothness of operation).
And as I said, I feel much better, having actually figured out the problem and fixed it. Not as dumb as I look, I guess.