A couple of comments, then I'll make some Austin recommendations:
Blue Bonnet Cafe is a great choice (I haven't tried the Little Diner.) Salt Lick is a great choice, and the sauce is the "star" there. In Lockhart, you can choose Kreuz, Smitty's, Black's, or Chisholm Trail. I still like Kreuz, but try their homemade sausage. (Order it with crackers instead of white bread, and get a Big Red; they'll think you're a "native".

)
Michael is right about the chili; although it's our official state dish, most of the good ol' chili parlors have gone by the wayside, so the best chili you can get is in someone's home. However, in Austin, you could try
Texas Chili Parlor on Lavaca.
A few places in Austin:
A "specialty" breakfast here is Migas or Huevos Rancheros. The best for both is
Joe's Bakery on east 7th. They also have great coffee.
Go to
Martin's KumBak (AKA "Dirty's") on Guadalupe for burgers, fries, and shakes/malts.
For chicken fried steak, go to
Hill's Cafe on south Congress. Many people also like
Threadgill's, north on Lamar (the original), or south at Riverside and 1st. Both have lots of good food to offer besides CFS, and both are old Austin institutions.
You have many choices for Mexican. I like
Guero's on south Congress,
Nuevo Leon on east 6th,
Jaime's on Red River,
Curra's on Oltorf,
El Patio on Guadalupe.
BBQ; the previous recs are good. In Austin,
Sam's BBQ is great, or drive to Taylor for
Louie Mueller's, considered by many to be the best in Texas.
There's an old steakhouse on west 6th named the
Hoffbrau. This is not affiliated at all with the national chain of the same name. Great for simple steaks, and they make a salad with a killer green olive dressing.
If you want to experience a real Texas dancehall, go to the
Broken Spoke on south Lamar.
Looks like you've already searched for the Texas restaurants from the homepage on this site, and I'm sure others will chime in. Have fun!