Very Large Array—Very Large Array is a premier astronomical radio observatory consisting of 27 radio antennas configured in a Y-shape, 50 miles west of Socorro, NM
Tombstone, AZ—Tombstone is silver-mining town made famous by the O.K. Corral gunfight.
Carlsbad Caverns, NM--Under the rocks, canyons and cactus are hidden 117 caves which formed when sulfuric acid dissolved the surrounding limestone.
Chaco Culture National Historical Park-The park is located in northwestern New Mexico, , in a relatively inaccessible canyon cut by the Chaco Wash. The canyon holds the most sweeping collection of ancient ruins north of Mexico.
Canyon de Chelly National Monument--Reflecting one of the longest continuously inhabited landscapes of North America, the cultural resources of Canyon de Chelly include distinctive architecture, artifacts, and rock imagery.
Monument Valley Navajo Tribal Park--This great valley boasts sandstone masterpieces that tower above the valley floor. Many familiar formations were featured in many Western movies.
El Camino Real International Heritage Center-- From 1598 until about 1885 colonists from Mexico and Spain entered New Mexico along El Camino Real, which extended from Mexico City and the port city of Veracruz in Mexico to Santa Fe, NM and beyond, a distance of over 1,500 miles.
Sandia Peak Tramway—This aerial tramway spans 2.7 miles of stunning natural beauty. From he observation deck atop 10,378 foot Sandia Peak an 11,000 square-mile panorama of the Rio Grande Valley unfolds.
I know there are other sights to see, but these are the ones that come to mind right now.