Depends on where you go. 13k' in Colorado in June can have snow, where in Phoenix it can be 113F. Late fall in out of the ordinary places like Bisbee, AZ, Bryce Canyon, UT, are awesome.
IMO, best tiems to travel:
Ouray/Silverton/Telluride/Montrose, CO - mid June to usually early November (All mountain passes closed here a few days ago due to early snow)
Pueblo, Denver, Loveland, Estes Park, CO - mid May to early December on average
Moab, Arches, Bryce, UT - Year 'round minus parts of spring. Mud gets pretty nasty at times
Flagstaff, Winslow, All of northern AZ - late March through Thanksgiving
Phoenix, Tucson, Bisbee, Douglas, Tombstone, Yuma, AZ - End of September to end of April. Bisbee/Tombstone areas a little later due to elevation.
IMO - Fall is the best time to travel anywhere in the US. However, fall for everyone is not the same. Wisconsin has most of their leaves on the ground already. Here in Texas, we wont lose our until close to Thanksgiving. Then again, we just had two days of light snow and lows in the low 20's the past 3 nights in West Texas. El paso even got snow, which is rare and even more so in October.