Hike
Gila Wilderness
The Gila Wilderness, near Silver City, is a great place for hiking and backpacking.
Organ Mountains
So are the Organ Mountains, near Las Cruces.
Gila Cliff Dwellings National Monument
Likewise the Gila Cliff Dwellings National Monument (http://www.nps.gov/gicl/), north of Silver City. Open year round for day use; fee $3/day for individuals, $10/day for vehicles, Park Pass applies. A pleasant short trail day use only leads to interesting archaeological sites. Hot springs in the backcountry are reachable via other trails. Camping is available at Gila National Forest campgrounds nearby.
El Morro National Monument
For the less ambitious hiker, El Morro National Monument (http://www.nps.gov/elmo), on the north edge of the region off NM 53, offers short hikes to a site of considerable historical and archaeological interest. Fee $3/day Park Pass applies; there is a small, decidedly primitive campground.
New Mexico Catwalk National Scenic Trail
another easy walk is 70 miles north of Silver City- New Mexico Catwalk National Scenic Trail. see: (http://travel.nytimes.com...)
Shakespeare
just south of Lordsburg is a curious ghost town with weekend tours with the family that owns it. Shakespeare (http://www.shakespearegho...) is one of those off the wall places worth a glance.
City of Rocks State Park
Off sr 61 south of silver city, has more hiking, through rock formations similar to those of the better-known and more extensive chiricahua national monument in arizona. small fee.
Rock Hound State Park
Near deming via sr 11, is that rarity, a state park where you are encouraged to remove parts of the park -- specifically, samples of the several unusual-but-not-precious minerals found there. fee $4/vehicle; each visitor may remove up to 15 pounds of rock. best at times other than summer, when it's beastly hot.
soak
When you're done hiking, go for a soak in a hot spring. The geologically recent volcanism that's widespread in this region has left a legacy of hydrothermal activity and a number of satisfactory hot springs, some developed, some in nearly pristine shape. Some places with hot springs in the region are:
Truth or Consequences -- formerly called "Hot Springs," in fact; several developed, commercial springs in town
San Francisco Hot Springs on the San Francisco River North of Silver City. Wild hot springs located in a designated day use only Forestry area. The springs are all underwater, lining the sides and bottom of the San Francisco River. Some people have stacked small rocks to form small pools where warm water can collect along the shores of the River. Reaching the springs requires a hike of a mile or two, and the trail often disappears along the way. Caution: dangerously pathogenic amoebae have been found in these wild springs, and in fact have caused a death or two. Make sure not to get spring water in your nose here good advice at any "wild" hot spring.
Faywood Hot Springs near City of Rocks. Lightly developed. Closed in 2008, following the death of the owner. The late owner's wife has listed Faywood Hot Springs for sale.
There are some other "wild" springs around the Gila Wilderness. Go exploring, and if you want some tips, a useful guidebook is Craig Martin's "Enchanted Waters: A Guide to New Mexico's Hot Springs," ISBN 0-87108-891-6. Note that this book is getting somewhat dated 1998, and ownership of developed springs, as well as access to back country ones, may have changed; inquire locally.