The River Walk, As a friend once said River Walk San Antonio is something everyone must do at least once in their lifetime. I also recommend the river boat ride. Boat drivers like to get tips at the end of the trip, and will make sure you have enjoyed your ride, you'll get to know quiet a bit of history, of the sites along the river. They will also add humor to keep you entertained.
Picture Below, of a restaurants with colorful umbrellas which shade riverside tables.
The Briscoe Western Art Museum is about history and culture of the American West.
If you have a Day to Spare, I do recommend SeaWorld San Antonio
Other interesting things to do are Joint Base San Antonio Air Show if you are town for one of there shows, its worth it.