Hit up Market Square durning the day and get your Mexican nick nack shopping done there, maybe get a Margarita from "Mi tierra" while your there. I suggest you get one of those rental bicycles to ride around in that way you get to see the most without all the walking. Go hit up southtown wich is over in the King William district, again you can ride the bike there, if you go there at night go visit Azuca and do some salsa dancing, great food there too BTW or go check out the Blue Star art complex. You can go to the tower of the americas and take a look on the observation deck there, you can also do all the typical toursist stuff like the Alamo, Seaworld, Fiesta Texas etc.
Addendum: Natural Bridge Caverns just north of S.A. is pretty cool too if you're into nature kind of stuff. I saw someone mention Lost Maples, but that would take way too long to travel if you're limited in time.