Silver City
The Palace Hotel, downtown, is a good place to stay.
Route 15
This is a beautiful road but is quite narrow by US standards. Can sometimes be closed due to snow in the winter. Spring and Fall are the best times although, due to the elevation, it's much cooler in the summer than the surrounding area.
Gila Cliff Dwellings
This National Park allows you to walk up and through ancient cliff dwellings.