A lot of it depends on what kind of racing you like, and how difficult you want the game to be.
For not-so-difficult games, pick up DiRT or Revo as mentioned previously. Another good alternative is to track down a copy of TOCA Race Driver 3 (US version). It's also known as V8 Race Driver 3 in Australia, and DTM Race Driver 3 in the Euro market.
It has a great mix of series, from 1950's open wheel cars to modern V8 Supercars, DTM, touring series, etc. Lots of fun to be had, and it's easy to learn. You'll be competitive in no time. Just be sure to use a wheel/pedal combo or a control pad w/ analog sticks.
Also, there's always the Need for Speed series. ProStreet is the latest iteration. NFS has undergone a major shift from its original focus on exotic supercars; with the "Underground" series it started focusing more on the "street tuner" scene. Not a big draw for me, I'd rather play a game where I'm racing a high powered Ferrari than a Honda Civic w/ stickers & spoiler. But with the last couple of NFS games they've started to mix up the car list a bit, and I have some friends who really enjoy ProStreet. I may have to pick it up at some point down the road.
Test Drive Unlimited is still available, but I'll warn you right now...it has a lot of bugs that remain unfixed and will probably never be looked at. Xbox360 players have gotten a fairly steady stream of new content, PC gamers have gotten almost zilch. The TDU PC forums are a grim, dark cavern, echoing with the desperate screams of long forgotten players hoping...praying...for something to keep them going.
On the simulation side, you have a number of alternatives. Here are a few that are available and two to keep an eye out for:
rFactor: The base game features fictional cars, as the game is meant for the modding community. This is the latest evolution of the ISI game engine (EA's F1 series, Sports Car GT, also used for SimBin's games). If you want a sim that provides a wealth of driving experiences, this is the game for you. Plenty of add-on cars and tracks available online for free.
GTR/GTR2: Focused on the world of sports car/endurance racing. GTR2 is a step up from the original and has a great "driving school" to get you familiar with how to drive. Also included in this series is GT Legends, featuring cars from years past.
Race '07: Primarily focused on touring cars, also includes some open wheelers. Great sim, and cheap. I saw it at Wal-Mart a couple of days ago for $20.
Live for Speed: Indie-developed racing sim, you can download a free demo but you have to pay for the full content. It's picking up a good following and has a pretty good reputation.
ARCA SimRacing: This one's still in development. Supposed to be a detailed stockcar simulator based on the ARCA series here in the US. Developers claim it'll blow Papyrus' NASCAR Racing 2003 out of the water, but that remains to be seen.
iRacing: In development, this is less of a game and supposedly more of an advanced simulator. The idea is to produce a simulator so accurate that real teams can use it for training & testing purposes. Known car types include short-track Late Model stock cars, open wheel cars from the Skip Barber racing school, as well as a few others. They will have varying monthly fees depending on what type of content you wish to access, and plan to have sponsored championships in the game. Their goals are lofty; they plan to create the most accurate racing simulation outside of what the big-budget Formula 1 teams use for their own in-house training & testing.