Well, Alpha Centauri is closer to Civ than GalCiv is, so I would suggest GalCiv. However, Alpha Centauri is a really,
really good game that drips with atmosphere. The old graphics may put you off if you're a graphics whore at all though.
GalCiv is also a brilliant game with really good AI. For a 4X it has particularly good diplomacy. You'd want to buy the expansions too because they really add to the game. I heartily recommend it and would probably rather play it than Alpha Centauri (just).
Other suggestions? Many people love Sword of the Stars+expansions; myself I couldn't get into it but lots and lots of people swear by it.
Space Empires 4 is probably the best turn-based sci-fi 4X I've ever played, although it is getting kind of dated now (not a problem for me). It's mind-bogglingly deep though, so it will take you a good while to learn.
Does it absolutely have to be turn-based? One of the best games I've played in the last few years is Sins of a Solar Empire, which is slow enough to almost be considered turn-based.

It's extremely polished, pretty, and incredibly fun. It never leaves my hard drive.
Star Ruler. Crazy game, this. Again it isn't turn-based but it is a 4X and the scale of the game is absolutely staggering. A nice feature is the built-in IRC client that connects to the game's channel so you can ask for help while playing.
