Get a tetanus shot first
Barton Springs Pool.
The Texas State Museum--excellent
The State Capital, make sure to walk to the back and downstairs to see where all the legislators are hidden underground so they didn't mess with the lines of the building.
Go to Wimberley--excellent sightseeing and antique shopping.
From Wimberley you can go over to San Marcos and see Aquarena Springs, one of the best and biggest Artesian Springs in the State. Bring your bathing suits and float down the river coming from the springs.
Make sure to tour the Hill Country by driving up the Colorado River to see Mansfield Dam, and Inks Lake