Lots of cool sites around San Antonio. There are Spanish missions much like what you have back in Cali, and of course the Alamo which is in the downtown area. In that same general area there is the Riverwalk with a bunch of high end stores, restaurants etc, and the Hemisfair tower from the 1968 World's Fair.
If you are more into outdoors activity, there is a Six Flags amusement park (Fiesta Texas) on the northwest side of town, right across the highway from UT San Antonio. It is very easy to get to from the airport area. A short drive futher in that same direction will take you to Boerne and then to Kerrville which are nice areas in the Hill Country - there are several caves like Cascade Caverns. Going a bit north from that area you can zip over to Fredericksburg (old German town with a nifty WWII museum), Luchenbach, Llano. Near Llano you can visit Enchanted Rock (big granite dome) or hunt for blue topaz, visit any of the Highland Lakes in the area.
Just north of San Antonio is New Braunfels, which has one of the world's top water parks (Schlitterbahn) centered on the Comal Springs, one of the biggest natural springs in Texas. In our state, clear natural ground water is an EXTREME rarity so the springs are kind a national treasure for folks. In this same area people go tubing on the Guadalupe River (floating along on an inner tube with a cooler of beer floating behind). A bit further north of New Braunfels is San Marcos, which has the former Aquarena Springs which is an even bigger spring than the one in New Braunfels, you can take a glass bottom boat ride or do some paddle boarding.
If you are a younger guy and looking for non-outdoors activity, about an hour north of San Antonio (easy drive on I-35) is Austin, home of the University of Texas and also the state's most "happening" night life venue which is the Sixth Street music district. Lots of fun, the general atmosphere in Austin would make a Californian feel right at home since the city is packed with expatriate Californians now. Austin of course has a major spring of its own, Barton Springs which is one of the biggest spring fed pools in the state and generally packed with a luscious assortment of UT co-eds that makes a a trip there well worthwhile.
If you are looking for more Texas-like activity, all around San Antonio are ranches and gun stores, you can go rent a horse and do some riding or visit a gun range and rent some hardware to do some target blasting, lots of fun stuff a very short drive from the airport area.
Hope you have a great trip!!!