While it straddles several genres, everything else in Spore just leads up to the part where it becomes a space game. The game is kind of hit or miss as far as people liking it though, and really don't have much story at all if that's what you're after. Still, worth checking out at least; you'll probably be able to figure out pretty well if it's something you'd like.
Someone mentioned Freelancer. That's a good one to start with as far as space sims go because it really works just fine with a mouse and keyboard, you don't need a joystick to play it well. It's got a decent story that gets a little weird sometimes, nothing special, but it works.
Tachyon: The Fringe is a lesser known space sim with quite a good story and a robust single-player experience, provided with a lot of delightfully snarky commentary due to the protagonist being voiced by Bruce Campbell of Evil Dead fame (who hams it up at times, but he hams it up so well that it's good anyway). It's a lot of fun and does have some replay value.
If you're willing to go very retro (like DOS/Windows 95 era), I played Privateer 2 (also without a joystick since I never got one to work on the same machine) many years ago and had a blast, or at least I seem to recall that I did. It does have a decent story. I keep wanting to play it but I never bother figuring out how to make it work on a modern PC. It uses live actors for cutscenes and while they may look a low-budget they don't look dated. When you're in space things do look dated, but since it's... well, space it's not that big of a deal.
Gratuitous Space Battles was also mentioned, and the title basically just says what it is. There are ships in space and they shoot lots of lasers and missiles and stuff at each other based on orders you have given them, and it's awesome. The story is basically nonexistent, with the backstory for each race just explaining why they want to blow up everybody else, but it's very fun and I believe there's a demo on Steam. There's no traditional multiplayer, instead players can post "challenges" where other players download one of their fleets to fight against. I'm not a big fan of MP either but I really enjoy this mechanic. It couldn't hurt to try the demo.