Among your first stops should be Barton Warnock Visitor Center. Here, you'll be able to orient yourself and get some great tips before you start exploring.
You'll soon find out why this Terlingua tourist information center is a hit with travelers. If you're looking for other things to see here, Mesa de Anguilla, and West Contrabando Trailhead are in the area too. They won't take very long to get to from Barton Warnock Visitor Center.