I'm going to do the Ruidoso trip this summer and try to see Outlaw (the LaFoy course mentioned), Rainmakers, and Inn of the Mountain Gods. If time permits I'd like to see Sierra Blanca again, but it will probably be three rounds in two days as it is.
If you fly into Albuquerque, Paa-Ko Ridge and Black Mesa are the two must see courses. If access works out, the Las Campanas courses are nice too, but not close to the top two. UNM Championship is very good, their other course is a little nine holer that continues to be squeezed by the university. Its a fun little nine to play, but wouldn't be a top recommendation.
Weather-wise--September/October is prime season generally. May--July is nice as long as you don't mind it getting hot, but its not like Phoenix. Temps depend on elevation, in Albuquerque we get a lot of days in the 90's. Ruidoso and Santa Fe are both at elevation as well. As Patrick discovered, April is NOT necessarily the month to visit--this week has been wonderful with 80 degree sunny days, but the winds kick up about 40% of the days and these are not golf-friendly winds.
EDIT: Oh yeah--forgot about August. That's the monsoon season. Great in the mornings, thunderstorms start around 2-4 p.m. pretty much every day, then nice evenings most of the time with a few lingering storms. Not a bad time to visit if you're a morning person.
In terms of other courses--Pinon Hills in Farmington is worth playing but well away from all the other courses mentioned. I like Pueblo de Cochiti as a shorter track that still captures some great New Mexico scenery. Some of the casino courses are fun--Santa Ana, Twin Warriors, and Sandia are all nice. I don't care for Isleta or Sonoma Ranch.
I've not played Albuquerque Country Club--its an older club by NM standards. Four Hills seems to be the best private club course in Albuquerque. ABQ CC is really short and tree-lined, but I've only seen it from the road.