Carlsbad Caverns National Park in southeastern New Mexico is the premiere cave system in the Southwest. The national park preserves the famous caves, which you can enter through either the natural entrance or by taking an elevator down to the main room. The park is in the Guadalupe Mountains, which continue into Texas to become the Guadalupe Mountains National Park. Many people visit both national parks on the same trip. In addition to the main cave, Carlsbad has several more adventurous caves to tour. You will need to sign up in advance for a tour led by a park ranger to go into these caves, and the tours can sell out quickly.
Another favorite Carlsbad activity is to watch the bats exit the mouth of the cave at night - this is seasonal, as the bats migrate to Mexico in the winter.
The park only offers backcountry camping, with no developed campgrounds. You can eat and stay in Whites City, which is a tourist trap of monumental proportions. Carlsbad, New Mexico is actually pretty far from the national park, but it is a nice, quiet town with lots of options. You could also camp at Guadalupe Mountains National Park and drive up to Carlsbad for the day.
Although it is not part of the park, you can visit Sitting Bull Falls in the national forest on a day trip - it is one of the most scenic waterfalls in the Southwest.