Race for the Galaxy is notoriously difficult to teach. Maybe not to motivated gamers, but the problem is that you need to sow the seeds for a few games before they hit that point where they just "get it" - seeing the possibilities with the game and start having fun.

Being one of the few people in my gaming group to enjoy and know Race, naturally I wanted to spread this. As with most games, the game really is quite a bit easier once you get going. To avoid being overwhelmed by the huge number of icons and wondered how they interact, and how someone moderately skillful (me) would dominate and demoralized, here is my approach. (This assumes we do a brief on the rules, with the emphasis on cost of a card and how to pay for it.)

Setup: I'm with 5 of my friends; I'm the only person that knows how to play.
Rules for game one:

1. No military worlds - everything is settled and paid for. The important part is going through the phases. On a second or third play, adding military to explain on phase 3 is simple once they understand the phases and the workflow of the game.
2. Chosen start world - I chose ELC for them. This is so at some point, you can teach "forced consumption".
3. No goals, takeovers, etc. - Probably obvious.
4. 2 players - I'm one player, all 4 of the friends are playing co-op. This way, if there are any questions, _everybody_ would pay attention, as this would affect all of them. Even better is that they learn all the icons at the same time, working out strategies together.
5. Go through every phase, every card on tableau - on every turn, we go through every phase - "did anyone choose phase one? did anyone choose phase two? etc". If someone did, we go through every card on the phase to ask if they have any powers. For dev/settle, we focus on getting the cost right, with the bonus.
6. Abbreviated game - we played to 6 tableau. At 3-4 tableau, once they start getting things going, I gave my board to control to a subset of them, and coached both sides. Tried (with some "cheating") to get every phase at least once to explain again (they will forget about it) what they do.

The benefit is that they get their hands on the cards and play with them, so the time spent is about the same as reading it all aloud.

As with Race, the best teaching strategy is an adaptive one! After the first shortened game, feel free to gradually add military, then goals. If you time it right, they're just "hey, by the way guys..." and they would adapt to it without a second thought! (Though there might be some groaning..)