See the Alamo if you must, but you should really get out of downtown and avoid downtown tourist traps. There are better restaurants and free parking outside of downtown.
McNay Art Museum is a must-see. The other museums are fine as well. Botanical Gardens are nice (and close to McNay) although likely not at their maximum lushness just yet.
The Toyota factory has a great tour by tram but you need reservations and no photography is allowed:
http://www.toyotatexas.com/tour/Tours/TourRes.aspx
The San Antonio Zoo is quite good.
If you like funky, there's Southtown and the area near King William that has little shops and art galleries and restaurants. Try Madhatter's Tea for sandwiches.
Or, if you're into history and serene 300-year-old buildings on acres of National Park System land, see one or two of the four Missions (other than the Alamo).
Comanche Lookout Park:
http://www.sanantonio.gov/parksandrec/directory_comanche_lookout.aspx
Barbecue: lots of random places, like Frankies (Fredericksburg, TX),
Salt Lick (Driftwood, TX), City Market and Luling Bar-B-Q (Luling, TX).
The Quadrangle at Fort Sam Houston would be a great place to shoot. There were animals wandering around inside last time I was there. However, call ahead because you may not be granted access to the base.
Government Canyon, a state park 35 minutes from downtown, has excellent hiking and mountain bike trails. Technical and non-technical.
Opening day for Texas League baseball is April 5.