I would go north or east. North gets you the Phoenix Sonoran Preserve (near I-17 and Dove Valley), Deem Hills, and Spur Cross. Look for lodging in Anthem, Cave Creek, Carefree, or far north Scottsdale.
Note that the area around and south of the I-17 and Loop 101 junction is NOT a safe area, but just north around I-17 and Pinnacle Peak or Happy Valley is fine and has several lodging options. Also, the areas around the Loop 101/AZ51 interchange and the northwest "corner" of Loop 101 (where it goes from an east/west to a north/south freeway in Peoria) are fine too.
North Phoenix also gives you an easy drive to Sedona, Prescott, and Flagstaff, which all have great hiking in the mountains.
East has the Superstition Mountains, McDowell Mountains, Four Peaks, and Salt River area. Look for hotels in Mesa, Fountain Hills, or Scottsdale. Chandler and Gilbert may also be viable, but are a little further away.
This also gives you good access via AZ87 (locally known as the Beeline Highway) to Payson and the Mogollon Rim.
Piestewa Peak (near AZ51 and Glendale Av) and Camelback Mountain (near Tatum and Lincoln) are pretty centrally located about 5 miles north of the airport. These are the two major urban mountains that everyone thinks about when they think hiking in Phoenix.
When will you be here?
edit: Here's a map with some of the more popular hiking areas circled:
