I would agree with TomD re UCLA (DD is currently attending). Since I am the designated Arizona Advocate, I can say that NAU would definitely meet your criterion. The trail heads on the San Francisco Peaks are easily within reach, although it is an uphill climb. Two National Monuments also are accessible (Sunset Crater and Walnut Canyon) and the Coconino Nat'l Forest boundary comes right down to town. Oak Creek Canyon and the Sycamore Canyon Wilderness Area is within reach. although it is a bit of a stretch on a bike. If you are a serious cyclist, even the Grand Canyon is possible. Lots of other options as you range further, many far off the beaten path, including many gems on Navajoland.

Much of the winter is not bicycling weather, but is more appropriate for skis or snowshoes; I have seen forty below temperatures in Flag on at least three occasions and I recall a monumental snowstorm in the late 60s where all access to the town was cut off by heavy snow for about two days. Snow has been recorded in every month of the year. The locals say there is nothing between Flag and the North Pole except a barbed wire fence (alternative saying -"nine months of winter and three months of hard sledding".

The greatest thing about Arizona is that you can find a good climate almost any time of year by simply adjusting your altitude. Flag is up at the cool end of the spectrum. It is heaven in the summer compared to the Dante's Inferno that is Phoenix.

I was stationed near Flag for three years (Wupatki Nat'l Mon) and enjoyed it immensely. More gory details on request..

Hope this adds up for you....