Tucson is surrounded by five mountain ranges:
The Catalinas, which are a high and large range, bound the north side of the city. The front-side faces Tucson and has peaks in the high-6,000 to 8,000 feet range. The northern part of the range is higher, topping out a 9,159 feet at Mount Lemmon. This is my favorite hiking area, as the trails generally have good length, significant vertical, spectacular views and good proximity to the city.
The Rincons, which bound the east side of the city. The Rincons have relatively high peaks, very long trails to the high country and very nice views. Unfortunately, they also have long flat sections at the bottom, which can be very hot in the warmer months. I enjoy the Douglas Springs to Mica Mountain trails quite a bit. These mountains also have good proximity to the city.
The Santa Ritas, which bound the south side of the city. The Santa Ritas have the highest peak in the Tucson region, Mount Wrightson (9,453 feet). The range is a bit farther from Tucson, about an hour south on I-19, in the Madera Canyon recreational area. The trails have lots of vertical and beautiful mainly-evergreen forests. The hiking is very well marked and relatively easy, considering the vertical. This area is very good for hot days in the desert, as the trails start at over 5,000 feet up. However, it can be very windy and cold on some of the ridges and at the top of Mount Wrightson.
The Tucson Mountains, which bound the west side of the city. These mountains are lower, with Wasson Peak providing the highest hiking, reaching 4,687 feet. These mountains have many trails, although all are less challenging than the longer Catalina, Rincon and Santa Rita trails. I have enjoyed the King Canyon, Hugh Norris and Sweetwater trails to the top of Wasson Peak. The views of the city are spectacular and the west side trails have incredible panoramas of saguaros and other desert plants. However, these trails can be quite hot in the warm months and they are shorter.
The Tortolita Mountains, which bound the northwest side of the northwestern Tucson exurbs, Oro Valley and Marana. These are a relatively small range with relatively low elevations. The highest point is 4,696 feet. The range has hiking, but the small range size and relatively low elevations means the hiking cannot approach the vertical or challenge of the Catalinas, Rincons or Santa Ritas. The Tortolitas are also relatively far from central Tucson.
The Huachuca Mountains, which are close to the Mexican border, are fairly far to the southeast of Tucson. I've included the range in the Tucson Mountains write-up, although it is a bit debatable whether they are Tucson-region mountains, as the drive is roughly 2.5 hours from Tucson, including about 1/2-3/4 of an hour on the very rough dirt and rock mountain road, W Montezuma Canyon Rd. The range is quite high, topping out at 9,466 feet on Miller Peak. The trails start quite high (above 5-6,000 feet), so the vegetation is typical of sky islands, including drought-resistent trees lower down and pines at higher altitudes. There are few cacti and none of the saguaros typical of the lower parts of the Tucson trails. It can be quite windy on the trails due to their high elevations. The area feels quite remote and wild, particularly if you use trails that start on the west side of the range as there are few signs of civilization in this area.