Not sure how much time you have, but for immediate gratification the west side is more accessible. My wife and I have hiked in Tucson for several years. Our favorite hike is Wrightson from
Madera Canyon south of Tucson. Just under 10,000 feet, a few good route options. We generally stick with the Old Baldy Trail (11 miles, about 4000 ft. elev. gain). Weather dependent (snow and ice can be deep).
To the north, our favorite hike is Mt. Kimball from
Sabino Canyon. We usually use the Finger Rock Trail (10 miles, 4000 elev. gain).
From the east, we love the Tanque Verde Ridge, which we've taken to Juniper Basin. (14 miles, about 3000 ft. elev. gain.) The east part of the park has a nice drive through several different desert areas; it's great if you can rent a bike.
These are (for us at least) all-day hikes that take you from the desert to "sky island" climates, so you see a wide variety of plants and wildlife.
In Saguaro Park West, Wasson is the main summit. Hugh Norris trail will get you there and back in under four hours, leaving plenty of time for the
Arizona–Sonora Desert Museum, which is worth a visit. Hugh Norris isn't very demanding; we like combining the Sweetwater Trail from Camino del Cerro, which meets up with the King Canyon Trail, and catch the Hugh Norris from there. Start early enough and part of that will be in the shade, plus it's a more diverse route than Hugh Norris.
We've been up to Mount Lemmon by car, it's an amazing drive but too long a hike for us to do in a day. However, there's a small community called Summerhaven up there which has a nice loop out into views of the Wilderness of Rock.
Happy to provide more info.