Drive to Big Bend National Park in West Texas. The area is very isolated, so you will likely spend most or all of a day (or more) getting there. If you are driving from the east on I-10, consider stopping to explore the Caverns of Sonora on the way.
When you arrive at the entrance to the park (Persimmon Gap Visitor Center), you still have a ways to drive. Make your way to the Chisos Basin and either camp at the Chisos Basin Campground or stay at the Chisos Mountains Lodge. Be advised that at least the campground only takes reservations six months in advance, and the reservations for March sell out immediately upon opening. You will probably have to stay up until midnight six months beforehand to book the campsite online. I would assume, but do not know for certain, that the Lodge is the same way. Either way, you will stay here for two nights. The restaurant in the Lodge is the only dining option, unless you cook your own food.
There are several good hiking trails available to you today, including the Window Trail and Big Mine Trails. If you feel up to it, you can even hike up Emory Peak, the most prominent in Texas. Afterwards, take a short drive east to check out other areas within the park, including the Boquillas Canyon Overlook, Rio Grande Village, and possibly another hiking trail, the Hot Spring Trail.
In the morning, finish up any hiking in Chisos Basin you want to, then pack up camp and head west to the iconic Santa Elena Canyon. The towering cliff walls are extremely dramatic and mark the border between the US and Mexico; as seen from the east, the right cliff is in the US and the left one is in Mexico. There is a short trail that goes partway into the canyon, although I recommend renting a canoe and paddling upsteam for a few hours. Then continue on to the revived ghost town of Terlingua, a former cinnabar mining town. You can camp at Living Rock Rentals or stay at Villa Terlingua. Dining options are limited in Terligua; Starlight Theatre Restaurant may be your best bet.