Since Texas is a ranching state, beef can be found everywhere. Steak is a contender for "most popular dish in Texas," and just about anything can be barbecued. Panhandle cuisine especially tends to capitalize on all those cattle herds and you can expect plenty a restaurant specializing in grilled, smoked, and savory meat and more meat.
Abilene is situated solidly in steak country and Abilene cuisine tends toward the thick and meaty, even beyond steak; roast, barbecue, ribs, and hamburgers. Amarillo is even more prominently a steak town. The famous Big Texan Steak Ranch is here and its 72-ounce steak. Barbecue also has a significant presence, meaty and beany as it is.
Of course, do keep in mind that all these cities serve other types of cuisine, too, of which Tex-Mex is prominent.