Out Old Spanish Trail to Saguaro National Park and around the one-way, one lane, nine-mile loop drive is a classic Tucson ride. To do it again is my only bucket-list goal.
Saguaro National Park head tube badge
Bike path in Tucson
You can continue out to Colossal Cave through similar landscape. As others have mentioned, there are miles of bike paths. Mt Lemmon, Sabino Canyon (time of day restricted now, I think), and over Gates Pass to the west are good rides. Even up little 'A' mountain is an interesting pedal. Most Tucson streets are wide, with generous shoulders. Some are way busier than others, of course. I'm slightly jealous.
My brother and I are hoping to visit Tucson again this next spring. Have fun!