If you've never been there, your first trip should go down the River Road, as it's incredibly beautiful. Have all your water and supplies. Take the second Cisco exit off I-70 going west (204, has a cell tower). It eventually drops into the river canyon. Camp at Fisher Towers and hike the trail, or if that's full, try Onion Creek or any of the river campgrounds. You used to be able to camp anywhere out there, but now it's pretty restricted, and you need a port a potty, even if it's just a bucket, if you're not in a developed campground with pit toilets, in case the rangers ask. After enjoying that country for a day or two, which could include driving up the Dolores River to the ford, Onion Creek, or taking the Dome Plateau road if you have 4X4, drive on into Moab for lunch at the Moab Diner or the Quesadilla Wagon, but be prepared for culture shock (tourists). You can then drive on out north of town for a quick visit to Arches or even go on up to the Island in the Sky, depending on how much time you have. The campgrounds in Arches will be full, but you might be able to get into Horsethief on the Island or even Dead Horse Park (too many people, but OK for a recon trip, but probably full anyway). There's some camping north of town off 191 though it's getting harder to find - one good spot is Little Valley, off the Klondike Bluffs Road (used to be called Burro Seep) out by the airport. Then go on out to I-70 and head home. That's a good intro to the area.