Time for my contribution, (I am an avid Trackmaniac!),
Trackmania United Forever is the best (it's the latest with everything and its the pay version and my third TM game), but if you don't find it anywhere, you can get the free version (It's not a demo like people think, its just one environment and limited features), Trackmania Nations Forever.
TMF (Collective term for both Trackmania United Forever and Trackamania Nations Forever), is quite cheap compared to any other game (well in Australia I've found, but it should be everywhere else), and its online/offline is so engaging, you'll find that you've missed a couple of days playing/creating/sharing.
With your computer specs, I suggest you try the earlier Trackmania games.
There are some other games made by Nadeo (people who made/developed the TM games), that are of the FPS category (Shootmania) and of the RPG category (Questmania), but they are yet to be released.