STAY
We stayed at the historic Menger Hotel directly across from the Alamo. I would not recommend staying there to be honest as it's pretty run down. However the location is amazing, and they have one of the oldest bars in Texas which is pretty cool! If I were to choose a different hotel downtown I'd recommend Hotel Havana or the Thompson. If your budget isn't an issue I would stay at Hotel Emma. It's the most beautiful hotel and centered in the historic Pearl District which has great shopping and restaurants. 
If you don't need to be downtown and you've got the family I recommend staying at the JW Marriott SA Hill Country which is in north San Antonio near Six Flags and Sea World. 

ACTIVITIES

The Alamo - In the center of downtown San Antonio you'll find the historic Alamo. Admission to the Alamo church where we walked around is free but other areas do require a ticket/reservation. It's open every day and certain days and times they have live reenactments.  

Witte Museum - One of the best science, nature and culture museums! It's the perfect indoor family activity if the weather is too hot or too cold. They have amazing exhibitions and while we visited there was a space/astronaut and shark exhibition. It's not cheap but you can definitely spend hours there! Another great kids museum is the DoSeum right down the street (similar to the Thinkery in Austin).

Historic Market Square - A place we love to visit is SA's market square! They have a few indoor shopping areas filled with dozens of individual artists and retailers selling trinkets, art, clothing, etc. The kids love walking into these stores! There's usually live music outside, and we always hit up Mi Tierra for a meal while we're down there.

If you extend your stay I also recommend a few other family focused activities including Six Flags, San Antonio Zoo, Sea World and Natural Bridge Caverns.

EAT

Best Quality Daughter - I have been wanting to dine at Best Quality Daughter for years, and it did not disappoint! It's family-style Asian-American cuisine and every dish was full of flavor. Not to mention the restaurant is located in a charming old home and decorated beautifully! It's located inside the historic Pearl District so a great location to walk around and shop after. 

La Panaderia Bakery & Cafe - Another San Antonio restaurant that has been on my foodie bucket list for years. I knew once I saw the line it would be worth the wait and it 100% was! They specialize in making handmade bread, pastries and pan dulce. We tried a bit of everything but the Ranchero Eggs Benedictine and Chorizo croissant breakfast sandwich were our favorites. It's counter service so my advice is get there early because the line does get quite long. 

Mi Tierra - If you spend time in the market area I HIGHLY recommend a visit to Mi Tierra for lunch or dinner. The decorations alone are worth it! They have a huge Mexican bakery and delicious margaritas. If you have an occasion to celebrate they have a live mariachi band inside you can pay to serenade your party! 

Riverwalk - We ate at Paesano's on the riverwalk, but I don't recommend. It's fun to walk along the riverwalk, but there are a lot of mediocre restaurants so do your research before choosing a restaurant!