Sabino Canyon - gorgeous and busy; as others suggested, avoid visiting the site during the weekend. The visitor center has maps for sale.
A drive up to Mount Lemon is totally awesome and the mountain is a welcome respite from the desert heat. Highly recommended as well for a hike, and changes of scenery along the 20 miles to the top are spectacular.
If you want to take a drive out of town, Mission San Xavier Del Bac, or White Dove of the Desert, is an authentic Spanish Baroque church that serves the local community. Don't be confused by distances displayed on the I-19 signs - the distances are in kilometers. Drive further South, get off the I-19 on Continental Road, and visit Madera Canyon; like Sabino Canyon, it costs a few dollars to visit, but... much less busy than Sabino Canyon, different scenery with a wonderful picnic area at the end of the road and several trails that lead up to the mountains from there.
If you are up to biking, rent a bike and take The Loop all the way around town. A great idea for avoiding car traffic and viewing local populace in its element.
Definitely try local food, the hole-in-the-wall places that sell Sonoran hot dogs are the best. Also try a raspado - like Hawaiian shaved ice, but better; real fruit in syrup, ice, condensed milk to top it off - delish!
I used to love Mount Graham for a couple of nights of camping. It has suffered a large wildfire last summer and I haven't been there since. The remote location makes it a nice place to get away from the crowds, though.
Chiricahua National Monument is a must-see if you have the time; the landscape of hoodoos is unique an worth getting to and it takes an hour or two on the trail to get to the most interesting rock formations. Off the beaten track in those parts is Fort Bowie, a historic site with ruins of an old fort.