A 512K broadband connection is just fine for online gaming. With a 1 Meg connection you dont really get any faster game play, but 1 meg is better for downloading. Dont go for anything less than 512K for online gaming, because you may be a little dissapointed.
If you have been playing on dial up, you will be truly amazed at how much better broadband is.
As for your system:-
A new graphics card might not go a miss. There are some real good bargains around at the moment. You can get TI4200 boards for about 120 to 130 quid (see Dabs.com) and these will work with just about any system. The newer Radeon 9500 and upwards series are a bit power hungry and you will have to check whether your power supply unit is up to it (ideally has to be over 300W).
What processor have you got?
If its a 1 ghz or faster a new graphics card and a broadband connection will improve your online gaming experience beyond your wildest dreams.
If you do get a new graphics card, like a TI4200 or something even better, make sure you crank up those video settings in your games to get the most out of it.