I don't know how long you are there, but I'll give you some thoughts.
For Mexican, I would do Rosarios, Mi Tierra or La Fonda. Rosario's is more hipster, but has my favorite salsa I've had anywhere. Mi Tierra is in the Mercado and it's been around forever. I think the food is still awesome though. It IS a little touristy though. La Fonda is another place that's been around forever. If the weather isn't too extreme, there is a great patio to eat on.
For breakfast you should go to magnolia pancake house at least once. It's excellent. Pancake Joe's is also great, but hard to get into. Donuts at the original donut shop.
For dinner, Boudro's on the river walk is still great. Bohanan's is a terrific steakhouse. Bella on the river is Italian/Mediterranean and excellent. Bliss is excellent and you'll need to make reservations way in advance.
Others - burgers at Chris Madrids, esquire tavern or the cove. Root beer at Schilo's (not a big fan of the food, but funny story, we were walking out of their, when we were down there for the Alamo. Robert Allen was walking down the street and I heard him tell whoever he was with that Les was going to lsu). Pizza - Dough pizzeria. They have a good Thai community there. I've been to a place that was great, but I can't remember it. Yelp.
If you want barbecue, stop in Austin and try Franklins or Labarbecue. I'd also recommend city market in Lulling. Unless something has changed, San Antonio barbecue is meh.
Have fun!