Among your first stops should be Panther Junction Visitor Center. Here, you'll be able to orient yourself and pick up some great ideas before you set out exploring.
You'll be spoiled for sightseeing choice when spending time in Big Bend National Park. Panther Junction Visitor Center is a great place to begin here, but there are loads of other worthwhile attractions to check out. Big Bend National Park, Panther Path Trail, and Sand Slide are close to the tourist information center and will fill in your day nicely.