Wing commander series.
Freelancer.
X series (if you're into BDSM).
Homeworld series.

Those should keep you occupied for a decade or so.

All genres? Obvious answer is Master of Orion :P

Wing Commander III, and TIE Fighter. Unfortunately TIE Fighter isn't on GOG. Freespace is one of the best, though. If you like RTS, try Haegemonia, or MOO. I suck at RTS, but I can't help loving them - sending hordes of tiny ships to their certain doom...

Space Rangers 2 is a gem you should check on if you haven't.
I love Freelancer personally, if you never played it, I think you can still find a hardcopy pretty cheap on the usual sites. It is still very fun to this day.
Star Trek: A Final Unity, was my favorite Star Trek game, though it was more adventure/story than space...I did like the space combat in that though too.
The Descent games here are very fun I think as well. :)
I'm sure there are many more, but those are my favorte space-like games that are on the top of my head.

There needs to be a "Everytime you mention Space Rangers 2, someone reinstalls it" meme, because I am one of those people who does it :P

That is what I am here for. ;-P
The funny part is, I have yet to buy the GOG version, as I have a hardcopy of Reboot that doesn't have any DRM once I put it on my machine. lol The game is so good though, I have been tempted to get the digital copy, but, common sense still has some small grasp on my sanity I haven't shooken free of... yet. ;-D

Here are some of the space-set games I've played that I think were among the best:

X-wing series (especially Tie Fighter) - if you can find them,
Homeworld 1 & 2,
Master of Orion 1 & 2 (on GOG),
Independence War 1 & 2 (on GOG)

There are more on GOG that I've gotten but haven't yet had the opportunity to try for myself. If you've finished Freespace have you played some of the user made mods and the Freespace Open Engine? If not, try the Blue Planet series. The first one Age of Aquarius is solid, but War in Heaven is fantastic.

I just won Dark Horizon in a contest here a few days ago, and I'm loving it. It not the best game ever made or anything, but it's a lot of fun :) It even got me searching all around town trying to buy a joystick, but I couldn't even find a shop that new what a joystick is :/

with obvious caveat to avoid Master of Orion 3 :)

If you are playing vanilla Freespace / Freespace 2 , be sure to investigate Freespace Open Source Project

Not only does it upgrade your Freespace experience with latest graphical whizzbangery, it also enables you to play many wonderful mods. For example, you can play the full Freespace 1 campaign in the Freespace 2 engine. And some of the Mod campaigns are as good and professional as the retail....

Other Space games of note...

Freelancer
Starlancer
Oolite
X-wing
Tie-Fighter
Tachyon
Wing Commander Series
Independence War Series
X series

Jets 'n' Guns is totally a space sim.
..or maybe not but I just want to recommend it, it's quite amazing.

Just read some reviews on it - if it's half as good as they say, I must have it too! But do you need part 1 to understand the story?

Also, Galactic Civilizations II is a nice game, if you can get past the there-is-no-DRM-but-if-you-want-patches-register-your-game-at-Impulse-DRM. Not quite as good as MOO2, I think, but it allows a lot more different strategies for victory. For example, you can spread your influence to dominate the galaxy culturally (colonies of other races will even join you, when your influence is strong enough).

And to answer the question that I would ask: There is not much backstory, so you don't need to know GalCiv 1.

The Independence war series is great. Highly recommended for anyone who loves more realistic space sims.
Wing Commander is a must play for anyone who likes space games. I just love the cheesy FMVs of the later entries in the series, they are great!
Emperor of the fading suns is a turnbased strategy game. Thing Master of orion meets Civilization, and they make a wonderful hybrid together. This one is tricky to find, and tricky to get to run on a modern OS (though it is possible). A true forgotten gem.
Sword of the stars. This is another 4X and a good entry level/relaxation game
X-series. This is not a series for those who don't have a lot of time on their hands. You are allowed to fly around as you wish, build your own commercial empire, fight battles where you control large fleets. This is one of the few real nonlinear games.
Tachyon is another great space flight sim with a nice setting and tight gameplay.

My personal favorites are games from Galactic Civilizations series. Both the original game (with its expansion) and the Galactic Civilziations II with all the expansions are absolute blast to play - immersive, great atmoshpere, just an awesome feeling of exploring the space and watching the planets turn around :)