If you consider weather alone, the most logical way to route your trip would be 1) Colorado, 2) Utah, 3) Arizona, and 4) New Mexico. That gets you out of the cool summer states before the first snow flies and to the warm (or HOT in the case of AZ) fall states in late September and early October.
Keep in mind that Trail Ridge Road in RMNP is nearly always closed by snow before October 1 (it is at 12,000 ft). The prime aspen viewing season in northern CO is the second full week in September, while from Chama, NM, to Durango, it is the last week of September.
Since you will primarily be concentrating on parks in southern Utah, I would recommend concentrating on parks and monuments in southern Colorado too. By limiting the miles you are driving, you can spend more time at Great Sand Dunes NP, Chimney Rock NM, the Durango & Silverton NG Railroad to Silverton, and Mesa Verde (don't miss the Anasazi Museum in Dolores). Then take CO 145 from Cortez or Dolores north to Telluride and Bridal Veil Falls, then CO 141 and CO 90 to Moab, UT, and Arches NP. If you are going to Capitol Reef NP, don't miss the incredible drive on UT 12 from there to Bryce Canyon NP.
For more ideas on what to see and potential campgrounds along this and other US highways in Colorado, see my new blog at https://2fanrgs.blogspot.com/