I live in San Antone, at least for now.
You'll want a plane, and you'll want good locks. Especially close to the border where planes occasionally "disappear".
Spike's right about Fredericksburg, great town. Good BBQ in Bandera, cowboy capital of the world, and home to Busbees BBQ. Austin is a lot of fun. Very GA unfriendly, though. As a university town, lots going on and a lot of eclectic stuff.
Then you have the Texas coast. The Gulf is like a giant toilet bowl (or so says a friend of mine), meaning that some of the dirt and trash from Houston and east finds its way around toward Corpus. Good shrimp, though, and some excellent birding along the coast.
San Antone is a party town, what with the Riverwalk and the turistas. There doesn't seem to be much in Del Rio... I'd guess you'd be traveling.
Destinations by plane: Mustang Beach, Burnet (airshow each spring), Fredericksburg, Houston, Galveston (walk to the air museum and Moody Gardens), and there's also a little private ranch strip at Kingsbury that has a fun little museum where they're restoring early vintage planes. You'll have to call, because I heard rumor that Kingsbury is only open to taildraggers.
http://www.vintageaviation.org/aerodrome/index.html