Barko, MarkBC and Highz all have good suggestions. The NP CG at Chiricahua is very nice and the trails that start there make for good hikes.
You might consider visiting Saguaro NP for interesting country, hiking and biking, however, no camping right in the park, though.
North of Tucson are the Santa Catalina Mts where you can find dispersed camping and good hiking near the top of Mt Lemmon. Depending, it might be snowy and cold but the road up the peak is a great bike ride. Lower some NF CGs might still be open. Sabino Canyon is lowest and close to town, with great hiking.
South of Tucson, just east of Green Valley you will find Madera Canyon, below Mt Wrightson, which has a nice little FS CG, hiking trails and mountain bike trails. Bring your binoculars if you like birds.
As Barko says, White Sands only has pack-in camping, but that too can be interesting. I just parked my All Terrain Camper and walked in 2 or 3 miles with a light pack and put my tent up in the middle of nowhere.
To the east of White Sands there is Oliver Lee State Park (showers!) and a tiny BLM CG at Three Rivers Petroglyphs and a large BLM CG at Valley of Fire NE of Carrizozo.
When do you start your trip, DD?