A unique place we visited was Organ Stop Pizza in Mesa. My sister and I were hesitant to go, as it was a favorite of the retirees who winter in Mesa, but it was actually very entertaining. The pizza was excellent and the pipe organ more impressive than expected. The organ player knew dozens of songs, and took requests. My personal favorite was when he played Under the Sea and they had bubbles as well as music filling the room!
The Mesa Southwest Museum was a very nice natural history museum, filled with Native American pottery, dinosaur bones, Old West artifacts like a replica jail, a stagecoach, and a place for panning for gold, and other unique artifacts like a suit of conquistador armor. We also visited the Roosevelt Dam, which, at the time it was completed in 1911, was the world's tallest masonry dam. There is also a rather interesting bridge just behind the dam and across the lake.
I loved the desert and scenery of Arizona, and my absolute favorite place that we visited was Tonto National Monument. The Salado Native Americans lived in the area between about 1150 A.D. and 1450 A.D., and they left some really fascinating cliff dwellings behind them. We visited the lower cliff dwellings, and the hike up to them (and view from them) were absolutely beautiful. We met a very nice park ranger, and she went with us to the top and gave us a little more information about the place. Combining beautiful views of the desert, hiking in excellent weather, history, and culture, this was definitely the culmination of everything I enjoyed about my time in Arizona.