Carolyn, there are a number of great summer camp options in Albuquerque.
The best resource I've seen is the free New Mexico Kids publication that you can pick up at various locations throughout the city. In terms of specific programs, UNM offers children's summer programs and so does Explora museum.
Good luck, and welcome to Albuquerque!
What kind of camps are you looking for? A week long sleep-over camp, a multi-week day camp?
In the latter category (although likely not in the "affordable" category), Sandia Prep offers their "Summer Prep" camp (http://www.sandiaprep.org/summerprep/index.aspx), and Albuquerque Academy offers their Th!nk Summer program (http://www.aa.edu/thinksummer). Both are June 4 - July 13, broken into two 3-week session. Both seem to run about $140 - $220 per class for each 3-week program, with about 8 classes filling a whole day (if you choose). The selection of classes is insanely awesome, and runs the gamut from traditional academics to arts, music, dance, theater, to sports and rec, technology, basic "play" classes for younger kids, etc. Both also have extended care to expand the full day from 7:30am to 5:30pm, if needed.
I have no personal experience with either of these, other than knowing that several of the teachers at our daycare make extra money by offering to stand in line (starting at something like 4am) for parents to register their kids -- apparently popular classes go quite quickly.
The JCC also appears to offer many special camps over the summer, some a week long (~$400), and a general summer camp that runs over five 2-week sessions ($500 for each 2 weeks). http://jccabq.org/main/camps.php
I think the zoo also offers some sort of summer program. If you belong to NM Sports and Wellness, they offer a summer day camp as well. You can find most of these pretty easily by doing a Google search for "Albuquerque summer camps", although I understand you're probably looking for personal recommendations...
Jen's advice about New Mexico Kids will give you the best overall reference for summer camps in Albuquerque. Additionally, I would also recommend Harwood Summer Art Camp, which now has a teen component (historically it has been 6 - 12 yos). They've got great instructors and a wide range of activities.
As you know the summer classes fill up quickly, the Albuquerque Museum has some great teachers teaching wonderful classes at reasonable prices. http://www.cabq.gov/museum/museumschool.html
NM Museum of Natural History & Science has great field camps for the ages of your kids -- starting the week of July 16. Art Adventures, Nature Journaling at each different habitat hike every day, is fun way to get to know all the cool zones around Burque. That one starts week of July 23.
-Cirrelda aka cc