Few tourist destinations in the continental United States are as inaccessible by air as Chama. No commercial airports are within a two hours' drive of the town -- Denver, Albuquerque, Pueblo, even the minor commuter terminals of Farmington and Durango, all are a considerable distance away. Train service is more remote still. You'll just about have to drive. The route from the south (US highway 84) is generally open year-round; approaching from Colorado via Colorado SR 17 and high Cumbres Pass can be problematic in the winter, although it's open more of the time than not.
The Cumbres and Toltec Scenic Railroad is one of the main attractions of the town. The railroad runs into the high country near Cumbres Pass, into scenic Toltec Gorge, and on to its opposite terminus at tiny Antonito, Colorado. Tour options range from half-way trips to and from a station near the Gorge (lunch provided -- not great dining) to round trips to Antonito, with return on either rail or bus.
Cumbres Pass becomes a wonderland for Nordic skiers and snowmobilers during the winter, owing to the high snowfall that this part of the Colorado Plateau usually receives. Some attempt has been made to separate high-country regions used by practitioners of the two sports, although conflicts still occur. Snowpack is variable year to year, but is usually satisfactory from December through February.
Viva Vera's Mexican Kitchen, on the south side of town, is good for New Mexican food (enchiladas, stuffed sopaipillas, etc.).