There are several commercial parks in the Moab area, but if you are boondocking, or want more scenery, Dead Horse State park, which has power (only) hookups is very nice. It is about 25 miles from town, up on the rim. There are also some even more isolated campgrounds in the Island in the Sky part of the National park, just beyond Dead Horse Start park.
Be sure to bring all the water you will need to use, little or none is available up on the mesa. It is all trucked in.
Of course Arches National Park is just out of Moab. The campground there is simply wonderful, hardly a bad site in the place. Get there early in the day.
It is a great time of the year to visit Moab, assuming no snow. It is a wonderful part of Utah to visit.
Green River State Park, up in Green River is also a good spot to stay, especially if you are getting into the area late in the day and don't want to drive down to Moab from the interstate.
Coming from the East, the Colorado National Monument just out of Grand Junction CO is very worth the stop, and has a very nice campground (no hookups, but a great spot to stay).