There are endless opportunities for hiking and riding out there.
For mountain rides: I would not miss the trail up the northwest side of the rincons. There are also some good trail on the back side of lemmon for this.
For road rides... well you are in road riding heaven. There is the santa catalina highway, numerous rides through the foothills on the south side west side of the catalinas. You can extend rides into the tucson mountains to the west head south and come around or head north and come around near marana. There is saguaro national park east and west. Be sure not to miss the desert museum.
First things first.. Stop in Summit Hut on Speedway and buy a topo of the Santa Catalinas for 6 bucks.
For hikes...
Mt Lemmon & Santa Catalinas: Sabino Canyon & Hutches Pool, Seven Falls & Bear Canyon, Palisades, Summit Rocks (lemmon rock overlook, you can easily hike to the top of the fortress & Ravens...PM me about this), Finger rock canyon to window (excellent) or you can make the bushwhack variation and do the fourth class scramble to the top of the rock guard (one of the best trips, PM me for details) , Romero pools from the Santa Catalina State Park, There are also many backpacking possibilities for the Santa Catalinas (more so than any other area mentioned, due to water availability)
La milagrosa canyon;
Rincons;
Madera Canyon to the summit of Mt. Wrightson (one of my favorites, santa rita range, talk to rangers before you set out... drug trafficking is common down there)
Baboquivari mountains are absolutly beautiful for hiking&climbing though jaguars have seemed to taken a liking to the area (second largest cat and the only one that hunts for fun).
The Dragoons are a great place to hike around -- through cochise stronghold up there.
Even more remote for tucson is the Chiracauas... Excellent hiking, it is a preserved area.
The santa theresa's are also remote but very dramatic.
Picacho Peak! may be a fun climb. You will see it if you drive from PHX to Tucson or vice versa.
Good Luck and Have Fun!