To be honest you are probably running an old motherboard and will therfore probably have an AGP slot and not a PCI slot for graphics.
You dont sound like you want to play the lastest and most demanding games... so a small amount of money could put you where you want to be. I suggest a graphics card upgrade (as your onboard graphics are rubish (no offence).
1st) - Check if you have AGP slot or PCI (or PCI-e) slot on your motherboard. This can be difficult to find out if you dont know the make/model of your motherboard (and of course if you couldnt tell the difference by looking). I suggest Belarc advisor 7. It is a little tool that examins your PC and tells you exactly whats what in your computer.
Use this to find your motherboard model/make. Then search google for that board - and find out what it takes (AGP or PCI ... you know)
SO... once you know this... get an appropriate card. I suggest for the mild gamer the 6200 or 6600 graphics. 6600Gt will play games like BF2 just fine... but your ram will hold you back from doing this... so to be honest you'd probably be best to upgrade that too.
Again.. is it SDram? DDR ram... and without even looking at your machine i can tell you its not DDR2 ram (lol). - Use belarc to tell you what you have. You want about a Gig so 2x512Mb would be good for you. Read up on the speeds of ram too - there are various sorts.
Ok.. thats enough from me... Hope it helps.