If you like to race bicycles then you're going to love Austin, Texas. It doesn't matter what time of year you visit there will be a race series covering the current season. I have met a lot of racers at the local races that were just visiting Austin from around Texas or from further away like California, Oregon, Utah, Arizona, Colorado, New Mexico, Missouri, and New York.
Don't travel with a bike? Don't worry! You can rent high-end race bikes for a number of Austin bikes shops (Ozone and Bicycle Sportshop) and you can also post to the many local bike forums for additional information and guidance. All you need is your kit, your helmet, and your shoes!
Keep in mind that the competition in Austin is steep. It is not unheard-of for Lance Armstrong to show up at a local race and there are several current and former national champions that live in Austin! However, races are always segregated by skill level and there are races for beginners as well as the pros!
Here is the spring/summer series for road races:
Here is the fall/winter series for cyclocross and MTB: