Personally, I really hate driving the Interstates. Highway 180 would be my choice, mainly because it is really a pretty drive, if the weather is good, which right now, it isn't. Timewise, I'm not sure which is best, but I try to make my trips as pleasant as possible and "cruise" instead of rush.
The Gila forest is around Silver City and lots of nice places there (Lake Roberts, Pinos Altos, and other nice areas). There is also City of Rocks, towards Deming and if you are into ghost towns, you can go to Shakespear, just outside of Lordsburg. There is a really good gunshop in Deming with a range if you want to shop or shoot. Hope this helps.
Have a good trip and drive safely.
"I don't care what everyone else shoots, as far as I'm concerned the .30-06 is God's caliber." C. Palmer