Can someone give a good summary of each expansion for Race for the Galaxy, and explain how it changes the game? Is it worth it to buy the expansions straight out? Or is it fine to stick with the base game as long as you want? What's the compelling reason to expand? Just curious!
The expansions all have different themes to them, and they change up the game play in significant ways. The first expansion, The Gathering Storm, is easily the most interesting in terms of changing up the game play. It adds in the goals, which channels play in different ways in each game. That, plus the new start worlds, provide enough extra variety that I'd suggest it's worth getting after playing roughly 25 games or so of the base game. That many games allows you to see the different strategies that are available and become familiar with the approaches.
Rebel vs. Imperium has an excellent new start world mechanic, where each player gets dealt two start worlds with their opening hand and then gets to make a choice. This, on its own, justifies the expansion I think. The takeover mechanic isn't especially interesting, it just doesn't come up terribly often, so really here, you're talking about just beefing up the deck and providing more variety.
The Brink of War, honestly, I don't care for. The prestige mechanics seem to tilt things pretty far back towards military being the most successful strategy. Perhaps I'm just not very adept at it, but after about 15 games or so, I'm close to taking the cards out of my deck.
Alien Artifacts is actually the start of a new expansion cycle. It's designed to be played with just the base game, and it changes things in a very different way than the first three. It introduces a map that the players build, with the players moving around on the map to find new stuff. Explore becomes a first-class action with the expansion, which isn't really much true in the other ones. I only have four or so games with it, and I think some of the rules for building the map are kind of inelegant (although understandable why they are the way they are), but it's a fun experience. I'd still probably buy it third of the four expansions, though.
The most compelling reason to get the RftG expansions is to get more cards overall. This makes the game more replayable and allows for different interactions between cards from the original set and the expansions.
As for each expansion, you can expect the following:
You can play RftG with all of these expansions by using Keldon's RftG AI.