You have chosen a beautiful place to visit! We own a rural lot there, and have explored the area enough for me to give you a few tips on camping.
There are a few RV campgrounds on the main drag (Hwy. 160), a couple east of town along the river (see AZ flycasters post) and a couple to the west. These would have hook-ups, etc. Pagosa is surrounded by National Forest, and I personally would send you to the camping areas near the end of Piedra Road, which goes due north from a mile or 2 west of downtown, up into the Weminuche Wilderness and San Juan Nat. Forest. Williams Reservior is up there, and there is some absolutely gorgeous, quiet, and peaceful boondocking in campgrounds, and un-official roadside pull-offs. This is a gravel road, well graded, and I'm thinking it's about a 25 mile drive from town. It is classic southern Colorado mountain scenery!! Again- no hook-ups, but I believe there are water taps and outhouses. Stock up on food, beverages, and firewood and head on up! And don't miss the hot springs, right in town- a bit pricey but beautiful and memorable, especially at night. PM me if you want more info- Tim