We have been living here in the San Antonio area for about 1.5 years now and love it. I have found a lot of info on the VisitSanAntonio website and tripadvisor.
Depending on when your spring break falls, it may be warm enough. Our Aquatica park will be opening in just a month, so we are eager for that as we became Seaworld AP holders last summer. Sea World is okay, we will go and let our daughter ride the roller coasters, maybe see a show. We enjoy the Aquatica park, except for the food. This Sea World has lousy food. So we bring in our own snacks and water now.
Six Flags we have avoided because we haven't enjoyed any SF park we have been to as adults.
People always talk about visiting the Alamo, but I would encourage you to visit all of the area Missions, depending on ages of any kids. There is a Mission Trail you can follow. Each has its own charms, and it is hard to pick a favorite. Most of our visitors have really enjoyed seeing Mission Espada and Mission San Jose. On Sundays many of them hold a Mariachi Mass.
There is a nice zoo at Brackenridge Park, as well as a train ride that circles the park.
Botanical Gardens are nice also.
Good places to shop abound. As mentioned by someone else, The Rim and La Cantera are nice shopping areas. We also like The Quarry.
There is a ton of construction on 1604 currently, so that may affect shopping in those areas. It's a nightmare currently.
The Witte Museum is a lot of fun. Free entry on Tuesday nights.
Right down the street is the McNay Museum, free entry on Thursday nights .
And the SA Museum of Art free on Tuesday nights.
Saturdays the Pearl Brewery Farmers Market is a happenin' spot, you can grab chicken and waffles for breakfast there.
Monster Mini Golf is lots of fun, we always take visiting family there. Indoor blacklight mini golf, and they play great music.
For food, we really enjoy Casa Rio on the Riverwalk. It is the original Mexican restaurant on the river, food and service have always been spot on. Some of the restuarants on the riverwalk are not so great.
At Market Square, we really like Mi Tierra. Be prepared to wait, but it is worth it. Market Square is a huge Mexican market, lots of street vendors around.
On Broadway, we like a burger place called Cheesy Janes. Great burgers and shakes, fun diner like feel in a restored gas station.
Another local burger chain is Chester's Hamburgers. Fantastic burgers, onion rings and shakes.
If you are up for a drive, off of I-10 a few miles off of 1604 is a bbq joint called Texas Pride.
http://www.texaspridebbq.net/ Great BBQ, and the line is always out the door. They also have a backyard area for kids to play in.
There are usually some good Groupons up for activities like mini golf, museums, and sometimes the attractions near the Alamo.
For New Braunfels, don't know if it will be warm enough to go tubing yet. If it is, there are tons of places to rent tubes. You can tube the comal, the guadalupe or San Marcos rivers.
We don't usually eat up that way though, so can't help to much with restaurants. Have heard great things about the Smokehouse though.