McDonald Observatory, Big Bend National Park
The heart of the Davis Mountains seems to be the ultimate spot for isolation away from city lights and polluted air, ideal for a state-of-the-art modern astronomical observatory. Run by the University of Texas at Austin, it has 3 telescopes used every night, weather permitting, by astronomers from all over the world.
During the day, you can drive up to the telescope coupolas for a free self-guided tour (9am-4pm). Guided tours are available for $7 per person at 11:30am and 2:30pm. They also have solar viewing during the day and star parties at night (go see their website below for more info).
Finally, the view from the top of this mountain is fantastic! Drive up there at least for that!!