You'll find Texas's most spectacular mountain scenery, as well as absolutely wonderful opportunities for hiking and other forms of outdoor recreation, at Big Bend and Guadalupe Mountains national parks. These parks also have an abundance of wildlife and both prehistoric and historic sites. Big Bend National Park is bounded by the Rio Grande, as it defines the U.S.-Mexico border, while Guadalupe Mountains National Park boasts the highest peak in Texas and a canyon that I believe has the prettiest scenery in the state, especially in the fall.
In addition to these two national parks in Texas, a third, Carlsbad Caverns National Park, is just over the state line in New Mexico. This easy side trip from Guadalupe Mountains National Park offers some of the world's most beautiful cave formations, and, if you're so inclined, the thrill of a true caving experience, as you crawl belly-to-rock through dirty, narrow, and dark underground passages.