The best possible way to get to Terlingua is surely a Southwest Airlines flight (catch connection from San Antonio, El Paso or Dallas) to Midland Airport. Then private shuttle to airstrip at Lajitas Resort. But if you are not rich like those with such options, I would consider a rental car from El Paso or Midland, TX. A rental car will also allow you to do the traveling necessary to see all you can. Big Bend National Park is about 1,500 square miles. The nearest large town is Alpine, 80 miles away. Lajitas is 13 miles away and many interesting sights are over 30 miles round trip. I don't believe there are any taxis available. There are Jeeps to rent in Terlingua Abaja, about four miles from Terlingua
Definitely a jeep. The main roads are all paved and the gravel roads are well-maintained. But there are so many interesting landmarks, ghost towns and villages that are somewhat difficult on a subcompact rental, plus you'll enjoy the open air.
You can get to west Texas by train, plain, and automobile, but the only way to get to Terlingua, Lajitas and the Big Bend is to drive.