I'll write a quick list geared towards history, culture, nature, authenticity over packaged tourist experiences. I would focus on the Southwestern area of the state.

>Rent a car with 4 wheel drive, fly into austin or san antonio
>Rent a bike and do mission reach trail, starting with Mission Concepcion and working down to Espada
>Visit Presidio La Bahia in Goliad
>cut across to San Ygnacio near the RGV, to see the Treviño-Uribe ranch
>keep driving north along the border, visit Republic of the Rio Grande Museum in Laredo and Los Regios Mexican Restaurant
>Drive north and away from border to Frio Town Cemetery outside Pearsall
>Up to Ruins of St Dominic Catholic Church in D'Hanis, and the D'Hanis Clay Products brick foundry
>Continue to Garner State Park
>Then drive to Rocksprings, visit historic town center
>Then from there drive to Devils River State Natural Area
>Then drive down to Del Rio and visit Cemetario Loma de la Cruz and Whitehead Memorial Museum
>Go west on highway 90 to Roy Bean Museum in Langtry
>Do the Cactus Capital hike in Sanderson
>in Marathon get BBQ and supplies from French Grocer
>Drive down to Big Bend and do hikes and/or camp in the Chisos Basin, other Big Bend stuff
>Leave by driving FM170 west along the border
>Outside Presidio, stop at Fort Leaton
>keep driving on FM170 for hours until you hit the town of Candelaria
>congrats you've reached the edge of the map
>turn around and go back to Rudiosa, check out the Sacred Heart Church
>Drive Pinto Canyon Road / FM2810 up to Marfa
>Continue on into Fort Davis
>Drive 118 up to McDonald Observatory
>Continue on and do Davis Mountains Preserve hike at the picnic area, or visit preserve if it's an open weekend (recommended)
>Continue up the road until you hit I10
>Take I10 east and exit for Balmorhea State Park in Toyahvale
>Take I10 east back to airport in Austin or San Antonio
>(optional last stop) Swim/hike at South Llano state park in Junction