I'd suggest staying within the Park; two nights on the east side at Rio Grande Village (the NPS CG, not the commercial one by the same name and next door) and one night Cottonwood. Three days is just enough to get a taste of Big Bend.
On the east side, the Hot Springs, Boquillas Canyon, RGV nature trail and a drive on the Old Ore Road (3-4 hrs) are a must. Ernst Tinaja off Old Ore Road is a great short hike. Go up to the Basin, hike the Window Trail or at least the Window View Trail and have dinner at the restaurant there. Lost Mine is a good hike there if you have time.
On the west side (Cottonwood), Ross Maxwell Scenic Drive, Santa Elena Canyon. Burro Mesa pouroff (short hike), take Old Maverick Road to western exit and have dinner at the Starlight Theater in Terlingua. La Kiva is an interesting local watering hole there also.