Most people that go to Big Bend National Park really only explore the Chisos Mountains or Santa Elena Canyon. Some may take a tour down the Rio Grande. But, Big Bend National Park is really a park best explored off of the beaten path.
Big Bend is about solitude and remoteness. You have the unique ability, unlike most places in the United States, to get far away from people and things. Not just have them out of sight, but to be miles and miles away.
Big Bend is best explored with a high clearance four wheel drive on its back roads. You can take the 50 mile River Road and do some tent camping down near the Rio Grande (just maybe not in the summertime, it gets hot by the river), or take the 26 mile Old Ore Road, if you truly want to go somewhere remote.
With your four wheel drive, you’ll be able to get access to some of the least traveled trails in the park, Like Strawhouse trail or Dominguez Trail. Don’t expect to see anybody out there while your hiking them.