In some ways, this is the best of times for space sim aficionados. FreeSpace 2, the last great commercial space combat sim is open-source and there are a lot of games based on that codebase which are freely available, or will be released shortly.
Personally I would recommend starting out with Freespace 2, picking an updated engine for it so it runs well on your system, and going to town. Then, if you’re interested, move on to all the other cool projects around the Freespace community.
Good Old Games also sells both Freespace 1 and Freespace 2 for $5.99 per game. They also sell a number of other fine space combat games. The game itself is excellent, it features the best capital ships in this kind of game that I’ve ever seen.
The biggest Freespace modding community appears to be at Hard Light Productions. They host a number of projects, though one of the most popular project out there is the Battlestar Galactica Total Conversion called Beyond the Red Line which has a demo available.
Hard Light does host a Babylon 5 Total Conversion called The Babylon Project, which seems like a good idea, but I haven’t tried it yet.
Unfortunately, no mod really seems to bring the graphics in the Freespace 2 Engine up to modern standards. Though it is certainly nicer looking than the wireframe graphics from the X-Wing games.
Older games can even be played through emulators if they don’t work natively. Ben Armstrong at Microsoft even made X-Wing run under Virtual PC. Unfortunately that solution won’t allow joysticks to work, so most people recommend using DosBox instead.
There is even a patch for the Win95 version of X-Wing to make it work on modern machines.
Newer games include the X series of space sims which are a bit more like Wing Commander Privateer and have you in a more open-space kind of profiteering game. Trading goods and ships for wealth and power respectively. These games are more like Elite and there are six in the series plus a stand-alone expansion here and there: