Here are some places we have enjoyed in AZ and some we have heard about and would want to see.
Arizona State Parks: http://www.pr.state.az.us/find/f_fac_camping.html
Dead Horse Ranch State Park
Homolovi Ruins State Park
I have heard Dead Horse is nice, it has been written about on the forums a lot.
Catalina State Park
Fool Hollow Lake Recreation Area
Lost Dutchman State Park
Lyman Lake State Park
Catalina is by Tucson & I have heard it is nice as is Lost Dutchman by Phoenix. Both of these have been written about a lot on the forums as well. Lyman Lake was nice, we stayed a night there years ago.
Kartchner Caverns State Park
Patagonia Lake State Park
Picacho Peak State Park
Roper Lake State Park
Kartchner Caverns, Patagonia Lake, Picacho Peak and Roper Lake should be nice ones. We spent a short night at Picacho Peak back in the early '70s and stopped to see the Saguros with our kids. We stopped at Roper Lake but did not stay the night in the late '90s, it was nice enough. Kartchner Caverns would be a good day or over night trip. Kartchner is on our to see list as is Patagonia.
These National Parks would be nice but would not have electric hook ups more than likely: http://www.nps.gov/state/AZ/
Canyon de Chelly by Chinle is great with a free campground and showers. Been there want to go again. http://www.nps.gov/cach/
Chiricahua National Monument by Willcox is great, been there will go back. http://www.nps.gov/chir/
Close by Chiricahua Nat'l Mon there is the neat ruins of a fort, some hiking to get to it. Fort Bowie National Historic Site is worth the 3 mile round-trip walk for the best experience of Fort Bowie National Historic Site. If you physically can not walk the trail, please contact the visitor center staff at 520-847-2500 for directions to the alternate access.
In the Bisbee Area: The town itself is neat to walk around in and we enjoyed the art galleries. The Queen Mine Tour was neat, we enjoyed it. Tombstone is very tourist trap site, if you get my meaning. The old Court House was interesting. Site of a lot of wild west history, but very commercial.
Near Wilcox is The Amerind Foundation ( http://en.wikipedia.org/wiki/Amerind_Foundation
) a ethnological, anthropological, archaeological museum and art gallery with paintings by 20th century Anglo and Native American artists. We enjoyed the art gallery and the museum. A neat out of the way, hidden site in the boulders of Texas Canyon. The Amerind is located in Cochise County, one mile south of Interstate 10, only about an hour east of Tucson, between Benson and Willcox. It is easy to find - just look for Dragoon Road exit, #318, and head south until you see mile marker 1 on one side of the road and the Amerind entrance on the other. http://www.amerind.org/
Cochise Strong Hold is in the same area, drove to it but did not camp, a nice setting. http://www.cochisestronghold.com/
In the Tucson area: Saguaro National Park near Tucson is great to see. Two sections seperated by Tucson. East and West. http://www.nps.gov/sagu/planyourvisit/directions.htm
No campground at either but there is camping very near the West as well as some not far on the East. We have never camped in them but we have visited the Nat'l Park both West and East.
The Arizona-Sonoran Desert Museum is really great (a zoo mainly without bars mostly) is over on the west side of Tucson by the Saguaro Nat'l Park and Old Tucson (the movie set/theme park). The Arizona-Sonoran Desert Museum is a must see for sure. http://www.desertmuseum.org/
Grand Canyon both sides are really nice, the south side by Flagstaff is close to Sunset Crater Volcano National Monument with camping nearby where we have stayed to see the area. (There is no camping in the monument. However, Bonito Campground, operated by the US Forest Service, is located across from the visitor center at Sunset Crater Volcano National Monument. This campground is generally open from late May through mid-October. There are no hook-ups.). There are nice campgrounds on the north side at the park and if full in the Nat'l Forest just before getting into the park. We stayed outside the park when there in summer and park cg was full. Grand Canyon http://www.nps.gov/grca/
Sunset Crater http://www.nps.gov/sucr/
Organ Pipe Cactus National Monument by Ajo has camping but is out in the outback for sure. Never been there. http://www.nps.gov/orpi/
Navajo National Monument is located at the end of State Highway 564 off of US Highway 160. We have not been to this one but sounds very interesting. Two campgrounds, one open all year, somewhat isolated but no fees are charged. This is on our to do list. http://www.nps.gov/nava/
Not far from Navajo Nat'l Mon't is the Monument Valley Navajo Tribal Park (aka Monument Valley Nat'l Park) entirely within the Navajo Indian Reservation on the Utah/Arizona border; the state line passes through the most famous landmarks, which are concentrated around the border near the small settlement of Goulding - this was established in 1923 as a trading post, and provides basic visitor services. We camped by the visitor center one summer, very monumental to say the least. A must see for any fans of old western films and some not so old ones as well.
Pipe Spring National Monument is 15 miles west of Fredonia, AZ. Off the beaten path, isolated with no camping at the Monument. But BLM land near by and other campgrounds. We have not been to this one.
BLM and other sites in Arizona web site:
Arizona has many Nat'l Forest campgrounds as well:
Apache-Sitgreaves National Forest http://www.fs.fed.us/r3/asnf/
We had a great time camping near Big Lake and canoeing on the lake. While canoe we watched an Osprey soop down and catch a fish and fly off toward the forest. The Big Lake Recreation Area offers four campgrounds, each designed for a different camping experience. Rainbow is the largest and most RV-friendly while Brookchar is located among a young stand of mixed pines and is designed with only walk-in tent sites. Cutthroat is also a tent-only campground with limited views of a large meadow and Big Lake. Grayling has both tent and RV-campsites among large pines. If I remember correctly we stayed at Rainbow. There was at the time a nice hiking/bike trail into the "bush" or backcountry. http://www.arizona-leisure.com/big-lake.html
Due to not having reseverations at the campground near Big Lake during a 4th of July, we spent some time near Show Low, camping I believe by Show Low Lake. I cannot remember the name of the site. It was by a lake and we canoe on it up to where the lake became a creek. We had a beaver slap it's tail at us which sounds a lot like a shot gun going off. We biked on side roads to the edge of the Mogollon Rim (pronounced: muggy-own) with trails down into it and perhaps camping in the area next to the hwy between Show Low and Pinetop-Lakeside. My kids were able to experience a small town 4th of July parade and fireworks show at the football field (see the score board catch on fire). We all had a blast. We also camped at another campground going back to Big Lake area off this same Hwy and dispearsed camped near Big Lake.
Coconino National Forest http://www.fs.fed.us/r3/coconino/
Coronado National Forest http://www.fs.fed.us/r3/coronado/
Kaibab National Forest http://www.fs.fed.us/r3/kai/
Prescott National Forest http://www.fs.fed.us/r3/prescott/
Tonto National Forest http://www.fs.fed.us/r3/tonto/home.shtml
We have stayed in a few. Very nice as our National Forests tend to be.
Here is a listing of AZ State Park Arizona Eileen recommends.