The Enchanted Circle scenic drive encompasses Taos, Angel Fire, Eagle Nest, Red River and Questa. You'll find a number of forest service campgrounds just outside of Taos towards Angel Fire and a couple more between Red River and Questa.
Of course, there are full service commercial campgrounds in nearly all of these towns.
Personally, though, I'd go to the Wild Rivers Recreation Area, now part of the BLM's brand new and large Rio Grande del Norte National Monument. Wild Rivers has a campground and is located several miles outside of Questa. There you can peer into the two gorges formed by the Red River and the Rio Grande. Hike until you drop, do some world-class fishing (after a good hike), study the petroglyphs, see wildlife all over the place.
:)
Lynn