Shortest route will likely not be the fastest because the long diagonals are on the US highway system going through towns rather than the Interstates that bypass them. There are several diagonals going SW through Kansas, W Texas and New Mexico. The one that looks good to me (because I've used most of the parts at one time or another) would be:
I-35 to Emporia
US-50 to west of Hutchison
K-61 to Pratt
US-54 to Alamagordo
US-70 to Las Cruces
I-10 into Phoenix.
Can't guarantee this is exactly the shortest, as there are three possible diagonals to the southwest breaking off of I-70 in Kansas, and I haven't added up the miles for the dozen or so possible combinations as those fan out. I just picked the one I've driven.
Likely the shortest driving time:
I-35 to Oklahoma City
I-44 to Wichita Falls
US-277 to Abilene
I-20 to I-10
I-10 to Phoenix.