This is the land of the open road. The best—and really—the only way to access the Hill Country is by car (or by motorcycle, which is an increasingly popular method). The towns are small enough that parking is not a problem. And though it's not encouraged, you're probably pretty safe leaving your car unlocked. Locals joke that they can spot an out-of-towner when they hear the "beep" of a car alarm.
You can rent a car from any of the major national chains at Austin Bergstrom International Airport and San Antonio International Airport.