Roger, We like the Red River area, in the northern part of the state (N36.708636° W105.40748°). It is in the Carson National Forest. There are a number of Forest Service campgrounds in the canyon to the west.
We will likely go to the
Columbine Campground sometime this summer/fall. Have not picked a date, yet, as we have so many other trips planned. We went to Red River a few years ago, but stayed in a small hotel. Wife liked the town, and the scenery is pretty-darn good. That's why we are going back.
Depending on how much time you allow, you could also hit Taos one day, or the Angel Fire or Eagles Nest areas. You could even fit a trip to the Sand Dunes (Colorado) into the trip, or make it a pre- or post-Red River stop (if you can get reservations).
Or, go to the Santa Fe area. There are many trailheads shown on the Forest Service maps to the northeast of town.
Enjoy your trip!