I stay in one of the cabins at Dead Horse State Park in Cottonwood. I take a road and a mtn. bike and ride from the park. Some of the road ride includes: Camp Verde and back, Page Springs and back and for mtn. biking some of the trails around the park and the Lime Kiln trail. Usually ride in the morning and hike in the afternoon, some mornings walk along the River Trail to Old Town Cottonwood for breakfeast.
Fools Hollow State Park in Showlow also offers some good road and mtn. biking as well though the mtn. biking on Timber Mesa is on the East end of town.
Kartchner Caverns is another State Park for road cycling when I have stayed there
None of the state parks are large enough to do road riding within the park unless you want to do laps but they are excellent as a base camp to ride from.