Second all that has been said. Georgetown (GTU) is the place for GA. Two main FBO's are Wright Aviation (930-7577,
www.wright-aviation.com) and Pilots Choice Aviation (869-1759,
http://www.pilotschoice.com/). My preference is Wright, they have newer planes such as an OMF Symphony-300, 2000 172R's with GPS (but more $) and friendly staff. PCA are cheaper and have a lot of older student 'beaters' (Tomahawks and Archers, no GPS ). Beth, the owner, can appear unfriendly, and the school does not allow night VFR X-C (strange!?!) but they are cheaper.
One thing about GTU is that it is a very busy uncontrolled airport, it is not unusual to have 4 in the cct and 3 at the hold. They have just agreed to install a control tower as they have had a number of 'problems' the most recent was a mid-air back in May. So keep your eyes outside and use the radio!
Finally other places worth going to are Fredericksburg (T82) for good sightseeing and Burnet (BMQ) for cct practice. Also for a good X-C go to either Waco or College Station and for a great day out go to Galveston (Moody Gardens and great Air Museum), but be up on Bravo airspace, as you'll either have to go through or around Houston.
Feel free to PM me for more details or if you want an intro to GA flying in Austin or want to do some flying with another pilot before launching off on your own.
- Tim