Since you are going to be in Arizona in winter, I'd suggest Sedona, which has many, many moderate hikes, and is less than two hours from Phoenix. Also, Tucson. I would suggest visiting Saguaro National Park West in Tucson. Also the Arizona-Sonora Desert Museum, which is the state's best introduction to itself, insofar as flora and fauna. AZSDM is kind of a zoo, not a stuffy museum, and is right next to the Saguaro National Park, which has many moderate trails. All these areas do attract many tourists, but they are also a very good intro to the state. And once you are out on the trail, it's the two-mile rule. Hike two miles or more from the trailhead, you'll lose most of the tourists.
You don't say if you are renting a high clearance vehicle or not. Many nice hikes in Sedona are to be found on very bumpy dirt roads, such as Secret Canyon Trail. There, I gave away a secret. Well, not really.
In Tucson, Catalina State Park and also Catalina Highway to Mt. Lemmon are both gateways to great hiking.
Flagstaff is beautiful, but often cold and snowy in March. Grand Canyon is worth a visit, and there are some neat hikes that are less popular that we can tell you about, but is about 4 hours or more north of Phoenix.
As for around Phoenix, I'll let the Phoenicians comment. They are stuck living in the megalopolis, the poor things, so they know the hikes where they can make a quick escape from the rat race.
