Definitely the River Walk.
The Alamo is one of five missions established in San Antonio back in the day. If you enjoy historical site (as I suspect you do), check out the other four missions!
http://visitsanantonio.com/visitors/play/history-heritage/san-antonio-missions/index.aspx
I second your suggestion. You will probably have a better time at the other 4 missions than at the Alamo.