Would help to know what systems you have...let's see...
Dreamcast:
Looney Tunes Space Race: Pretty much a Mario Kart game, but the Looney Tunes style really shines through, especially with the cel-shaded style and weapons like the Extendable Boxing Glove, Portable Hole, Thundercloud-in-a-Can, and of course falling Anvils. Great four-player fun, even if you lose.
Starlancer: Imagine if Ace Combat was set in space, and this is what you'll get. Admittedly, the learning curve is absurdly steep, but the sheer amount of control you have piloting your ship is great. Also a great atmosphere, with a nice HUD, plenty of comm chatter, and a nice little story about your rise through the ranks.
PS2:
007: Everything or Nothing: Cover-based third-person shooter, but my sibling and I still have a lot of fun with this game, admittedly because the co-op can be crazy fun...and funny. One time we decided to take out the first guard in one level by chucking seashells at him. And the stealth sections are a bit difficult due to the one-stick-movement-scheme, but getting through them just made us feel even more awesome.