Ruidoso is a mountain resort town about 1h from Alamogordo. It lies on 6.920 ft in the rugged Sierra Blanca Mountain range. You'll find a lot of nice shops and restaurants there.

If you like winter sports or hiking you should go 10 miles more up and visit the countries southernmost ski area "Ski Apache". It's owned by the Mescalero Apache Tribe. From Ski Apache you can hike to the Sierra Blanca Peak (11.981ft), also called White Mountains, the seventh highest peak in the state. After the trip you could stop for a break at the "Inn of the Mountain Gods" to play in the casino or enjoy a dinner with a view to the lake. 

https://www.ruidoso.net/      https://www.skiapache.com/