I think JoJo gave you some great ideas. I love Houston, but mainly because I have a bunch of friends there and we go to many bars and do a lot of shopping there. But for the purposes of your trip, I think you could skip it. It will be hot and humid, but at least you're coming in June and not August.
The various Theme Parks will probably be fun for your family, so make sure to check out Sea World or Fiesta Texas.
The hill country is a pretty area, and there are lots of little towns that have antiques and other eclectic things. Of course my dad calls it all "junk", but I know a lot of people just like to go from town to town and look at knick knacks and other unique things to the area. Some town ideas are Fredericksburg, Boerne, Wimberley, Gruene (which is kind of between San Antonio and Austin), and Marble Falls.
I agree with the Port Aransas comment as well. If you do go to the beach in the Corpus Christi area, make sure it is on Padre Island and NOT the Corpus Christi Bay, which is just over the bridge from downtown. Padre Island is very, very long so I don't mean "South Padre" which is a couple hours south of Corpus. I mean "North Padre" or Mustang Island, which is a few minutes from Corpus. Port A is also in this area.