Our first trip to Arizona was last October and we spent most of it in Tucson. The University of Arizona is a pretty place to stroll around, and there is a section with a lot of eating places and shops that we enjoyed checking out. There are also museums on campus, (Arizona State Museum, Arizona Historical Society, Center for Creative Photography, and others), that might interest you.
Like other posters have mentioned, the Arizona-Sonora Desert Museum is a great place to spend some time. I'm not sure why it's called a museum because most of the animals, (we had never seen javalinas before!), and sites are outside. Be prepared to walk a lot! Wear comfortable shoes and don't forget the sunscreen. We were there the end of October and I couldn't believe the intensity of the sun- wow! My husband reminded me that yes, it's hot, it's the desert!

Also, be sure not to feed the wild coyotes! We got a kick out of the signs near the parking lot.
The drive to and from the Sonora Desert was amazingly beautiful. The road wound up into the mountains and there were scenic sites to stop at where picture taking was awesome. Lots of cacti, which we really couldn't take our eyes off. Coming from New England, it all looked very strange! It was also fun spotting the very expensive houses that dotted the landscape and seemed to be built on the edges of cliffs.
If you like to gamble, there are a couple of casinos in the area. We spent some time at one that wasn't too far from the Tucson airport. The casinos aren't huge, but fun if you like that sort of thing. While my husband won on Black Jack, I lost on the slots. Easy come, easy go!
We also spent some time in the Phoenix/Mesa area, which is about a two hour drive from Tucson.