The Carson National Forest has an incredible number of trails suitable for day hiking and long-distance backpacking.  It would be impractical to list and describe all of them.  So instead, here is a link to the Carson National Forest website:

https://www.fs.usda.gov/activity/carson/recreation/hiking

List of mountain biking trails:

https://www.fs.usda.gov/activity/carson/recreation/bicycling/?recid=44054&actid=24

List of equestrian trails:

https://www.fs.usda.gov/activity/carson/recreation/horseriding-camping/?recid=44054&actid=104

Forest maps and more information are available from the Forest Service.  The headquarters for the Carson National Forest is in Taos, and the nearest Ranger District office is located in Questa.