In the Spring start south and work your way north. In the Fall start north and work your way south. The two trips we went to NM and AZ we left home first week of May. Still very hot in southern NM at Roswell - 105 degrees.
When we go to Yellowstone & Grand Teton, we try to get there by the beginning of June to beat the crowds.
Also, 3 weeks isn't a lot of time to do a lot of sightseeing and make a southern to northern loop (or vice versa). To make driving that much distance worthwhile, we go for more like 8 weeks. It's a long drive across the country.