GRID, Burnout Paradise, and Midnight Club LA are my suggestions.
Start with Burnout Paradise. It's got an absolute shitload of content, it's polished, the visuals are nice, it's fast, it controls well, etc etc it's a cool game that should be supported by cool people. Oh and its only twenty bucks, which is really great considering how good it looks and just how big the game is. You can buy it off of the Playstation Network or at retail.
Yeah Burnout Paradise has got loads of gameplay in it more when you play ONLINE
Don't buy the the ULITMATE BOX set its a pointless overspend as the normal Burnout Paradise site gives you all you need by downloading the content that's comes in the Ultimate set such as the motorbikes addon and an update that you get free when you connect to online play
Gran Turismo 5 Prologue is also an awesome game offline and online play ... if you played pacific rift then grab the older MOTOSTORM its still holds me for gameplay and its damn cheap pre owned too!
Dirt is ok but I prefer GT5 over it for realism!
I've played F1 but found it pretty sucky
Out of all the driving games I always go back to playing Burnout Paradise and causing mayhem in online matches when they break out into games Tag when I wreck their car and they try catch up with me to do the same all hell breaks loose lol
Need For Speed: Shift is coming out, looks pretty bad ass (Even though I'm not one for racing games, gameplay footage looks awesome). Burnout Paradise is a great game no matter if you like racing games or not.
to be honest its not that great unless you have several people at your house all wanting to play pass the gamepad around fine if your into that kind of gaming but pointless if your not playing with a party of friends at your house!