A nice approximately 60 minute drive northeast on U.S. 70 will take you to White Sands national park and then on to Alamogordo. From there you can continue up to Cloudcroft or drive through scenic Tularosa and on to Ruidoso. Even if you don't gamble, the Inn of the Mountain Gods is worth a stop for the gorgeous views from the panoramic window.
Stop at Eagle Ranch just outside Alamogordo for pistachios, wine and ice cream.
A nice southeasterly drive down Highway 28 will take you through Old Mesilla - you'll want to stop here - and then through acres of pecan trees. Stop at Stahmann's for great nuts and chocolate.
East on I-10 takes you to El Paso, west to Deming, Silver City...
North on I-25 takes you to Truth or Consequences and Elephant Butte...
If you are feeling really ambitious, a three hour drive will get you to Albuquerque - lots to see and do there.
A few ideas here:
Scenic Drives near Las Cruces, New Mexico