Don't mess with the Alamo
As a non-native, but long-time (former) resident of TX... Don't mess with the Alamo! It's important!
Seriously, San Antonio is a very fun city. Great climate, wonderful people, very casual and comfortable attitude, great food, great drinks, etc. The Riverwalk is excellent. One of my favorite urban venues. If you are staying close enough to walk you can really have a great time. Don't worry about the best place to go. Everywhere will be filled with tourists. Most of them will be tourists from other parts of TX. Lots of them will be looking for excitement in San Antonio. Just enjoy the Riverwalk and stop at any place that looks appealing to you. I doubt it will disappoint.
It will be a markedly different experience from NYC.