Unfortunately, there are few routes available through the mountains. I found Route 8 east from San Diego to Route 10 between Phoenix and Tucson pretty neat. You avoid Phoenix. But I've always wanted to drive through Ajo (Route 86 from Gila Bend to Tucson). But once you get to Benson, AZ, go south and go through Tombstone (OK Corral) and Bisbee AZ (art town) then Route 80 back to Route 10 through Lordsburg (ghost town).
This just gets you into the state. If you want to continue crossing it, you can take Route 10 into Las Cruces and then take Route 70 through White Sands to Route 82 to Cloudcroft (neat skiing/artsy town) and then head south to Carlsbad Caverns. At that point, you are almost to Texas.