Myself, I wouldn't put any money into it. Put that money towards a new computer what ever it would be. ... Use it the way it is until you pull your hair out.
Well, that's kind of silly. Why use a computer that will be painfully slow for no good reason when even $40 (for 1GB RAM) would make it a drastically more pleasant experience, not to mention extend its useful life?
And really, a 1.8GHz G5 with a gig or so of RAM is a comfortably usable computer for what a lot of people do. My work machine is a measly 2x1GHz G4 and it's not like I'm hurting when doing anything short of video encoding, and we have people using the slower version of that same iMac. Certainly for what the original poster mentioned--playing card games and surfing the web--it's perfectly adequate.
As for the OS, it shipped with 10.3. If you've never updated it, the best browser you're going to have access to is Camino (Firefox 3, Safari 4, and the latest Opera all require at least 10.4). It's actually the browser I use most, but Safari 4 is going to be noticeably faster (again, with enough RAM--seriously, get some), so updating to 10.4 or 10.5 would be worth the expense, if you ask me.
You can check what you've got installed by selecting "About This Mac" from the Apple menu.
Also note that, if the computer does up and die on you, the RAM and OS you bought for it should still maintain some resale value. Personally, I'd beef it up a bit and enjoy it as a backup machine, then when you eventually upgrade or stop using it, sell it or donate it to someone.