Coming to Houston on Biz
I agree that Cafe Annie is a good choice, and would also like to suggest Marks even though it's not completely Texan fare - but amazing food and you wouldn't be dissappointed. Another place is The Rainbow Lodge, which is extremely decadent and specialize in wild and exotic meats, but also have amazing seafood dishes and an incredible atmosphere. It's the kind of place you want to sit around and linger for hours and hours. They both serve amazing dishes and have a goreous wine list. If you want something really upscale I have heard that the Capital Grill is also on top of the list, but I have not personally been but have heard very positive reviews.
For Brunch I would highly recommend the Backstreet Cafe - it is lovely and in a great part of town, and the dinner is great but the Brunch is fabulous and so much fun. It is a local favorite. It's a little more upscale but so much fun and still casual but sophisticated. Another brunch spot is Rio Ranch, and that is more down-home but very good.
I'd say Cafe Annie for fare that is unique to Texas and upscale, everything else that says Texas is not upscale. Brunch would be good at Hugo's (even though it is a buffet - I know, I know) it is nicer/innovative Mexican food, great drinks, just get there before 12 and beat the rush.
As for Tex Mex/Mexican or BBQ, everyone has strong opinions, if you had time at night for fun and food try Armadillo Palace which has live music, people dancing and things like chicken fried steak and across the street is Goode Company BBQ, but neither are divey places that people tend to picture in their head when they think of Texas casual.