I too would reccomend hitting Canyon de Chelly, especially if you are inclined to take a jeep or horseback tour.
If you hit Las Cruces, I would definitely recommend heading to White Sands National Monument and also going to Old Mesilla (which if you are at all interested in US/Mexico history was once in the demilitarized zone between the two following the Mexican/American War). El Paso (in TX but just 30 or so miles from Las Cruces) can get a little sketchy since it is a border town, but if you stay on the NM side they have some decent western wear and other outlets for those sort of goods.
If you have time and like art or culture you can't beat Santa Fe. That is where the Georgia O'Keeffe museum is, and her home is on the way to Santa Fe from Alburquerque. There is also the Palacio del Gobernadors in Santa Fe where the local Pueblo people still sell their artwork and jewelry open air like they did in the 1600s. Their is also the orchestra/opera in Santa Fe. Taos is a huge art community with some more Billy the Kid history. You might also be interested in touring Taos or Acoma pueblo. Taos competes with another pueblo for being the longest continuously occupied communities in North America and are quite beautiful.
Carlsbad caverns is magnificent as well.
If you are interested in visiting the Navajo reservation I would recommend Canyon de Chelly. It is a great look at Navajo life prior to some of the disruptive affects of colonization. Window rock is also nice, it is the seat of government for the Navajo Nation. Navajo culture is incredibly rich, but the Navajoland is incredibly impoverished.
I really like the Rack in Phoenix, which also has the Hurd museum. Another great place is the Petrified Forest in AZ, or El Malpais in NM if you are looking for more natural wonders. If you head down south, make sure to see some saguaros.
Mesa Verde is a super place to go, although in October you might hit snow. If you need more info about this one just PM me-- I used to be an employee and am a local, so I am well-informed.