My understanding is that you just need to keep doing mission for Federation groups. The best ones with the best gains are where you bring them something for free. They rank you up faster. At a certain point when you do enough, you'll get offered a progression mission.
What I've been doing is finding a nice location to setup for missions (good mix of economies nearby), then rank that station up to Allied by doing mission mostly for the faction that owns the station and is in the Federation. By the time I get Allied, I usually get a progression mission too. Then, I move on. You can't go higher than Allied with a group. I'm up to Warrant Officer with Federation now.
I actually started doing this with Empire now too. It seems to be about the same.
I mostly do trade missions since the combat ones are a bit buggy and just take a while to do all the USSs. I believe bounty hunting and even just trading goods also help. But, the missions are better, especially the charity ones.