Keep in mind that it's a small town of 25,000 to 30,000 so not alot of options but i found you some info and photo's (Not mine). Very hardcore desert (Chihuahuan) and there is the Pecos River that flows thru town and Brantley Reservoir just north in which you can Water/Jet Ski.
Carlsbad Parks and Rec site. Nice web page.
Carlsbad, NM: Official City Web Site - Parks and Rec Dept (http://www.cityofcarlsbadnm.com/parks_and_recreation/index.html - broken link)
Carlsbad area basin
Looking west to the Guadalupe Mountains
Guadalupe Mountains National Park just across the Texas Border