Epik

Best answer as chosen by Asker / Ratings
First off, you must have an active working Internet connection plugged in to your Xbox 360 (Ethernet or the Wireless Adapter which can be purchased for about $20 USD); Then, go to "Custom Match" and choose the variables for a race, such as Sprint, Circuit, Speedbreaker, etc.

After you've set up the race type, press the Start button and you will start searching for either a match or players to join your host game. For new players, I suggest playing unranked, but there usually aren't that many people playing on unranked, so Ranked could possibly be a good way to get started, depending on your skill.