River Walk and Alamo are close to each other, both sits at the center of downtown.
River Walk is really a river for you to talk a walk. You can walk and take a boat ride. There are many restaurants along the river. It takes about 4 hours. River Walk is probably the best you can see in TX. Don't miss the boat ride and Mexico food there.
Alamo is shown in a building complex which was a fighting place for TX's indenpendence from Mexico. Prepare 2 hrs if you like history, otherwise, half hr.
On the northeast surburban of San Antonio, the natural bridge cavern is one of the most beautiful cavern in US. You may want to take a look if time permits.
Have fun!