Red River's main activities are outdoors: skiing and tubing in winter, hiking, fishing, biking, and OHV routes once the snow melts. The town is small enough that Main Street and the riverfront parks are part of the experience, especially if you are travelling with children.
Use Red River as a base rather than a checklist. One good day might mean the ski area in the morning, a slow walk through town, and a drive over Bobcat Pass or around the Enchanted Circle before sunset.