If you can do without hookups, I'd suggest spending a few days at Rio Grande Village (the NPS CG and not the concession operated one), exploring that area and then moving to Cottonwood on the western side to check out things on that end.
First, get a decent map of the Park. The National Geographic version sold there will work unless you're planning on off trail bushwhacking.
Second, plan your visit. The Park newspaper usually has great suggestions for things to do and see.
The Paisano Third, check with the rangers for current road conditions. The back roads in Big Bend lead to many really nice points of interest but aren't heavily travelled and getting stranded isn't good.
We've driven all the roads, done most of the hikes and enjoyed them all. Not a single one that we wouldn't recommend.
Outside the Park on the west end, La Kiva is a unique watering hole and the Starlight Theater has unexpectedly good food and entertainment.
Enjoy your visit and post back with your experiences.