San Pedro Parks Wilderness is a significant hiking/backpacking attraction in Santa Fe National Forest, with its most accessible trailheads reached via SR 126 east of town. Bring rainwear and wear water-resistant boots; this area gets far more precipitation than you'd expect in New Mexico, and the trails are frequently muddy if not downright submerged. Pleasant scenery, with rolling hills and high mountain "parks" (open, grassy valleys).
The beautiful Jemez Mountains lie beyond San Pedro Parks (which isn't part of the Jemez, but rather the lower, less prominent Nacimiento Mountains), with numerous points of interest. Warning: SR 126, although a "state highway," gets nasty east of San Pedro Parks. A bucolic-type vehicle is a good idea if there has been recent rain, and the road may be closed following snowstorms or other bad weather. If you're driving a standard sedan, you should be OK in dry weather, but the road has some exposed spots with deep ruts and is not for the faint of heart.