I just did pretty much the route you are describing as far as Dodge City.
Journal with map at:
http://www.crazyguyonabike.com/doc/5841I went ST to Globe then north to Holbrook to catch Rte 66. Rte 66 to Santa Fe then caught the Santa Fe Trail to Dodge City.
I don't recommend the way I did it because you have to cross the Salt River Canyon. You can head north out of Phoenix but you can't ride the interstate so you will have to go through Sedona and then climb out Oak Creek Canyon (tough) to get to Flagstaff.
If I were to do it again, I would either take the ST to Blythe then follow the Colorado River north to Kingman then jump on I-40 to Flagstaff or I would take ST all the way to I-25 in New Mexico then head north.
I would prefer the first option mainly because I really enjoyed the section of Rte 66 from Gallup to Albequerque. I also really liked the Santa Fe Trail.
The route from Gallup to Dodge City was better than anything I've seen on the ST.