The high mountains pose the usual mountain hazards altitude sickness and avalanche danger for the hiker/skier, snowpacked roads in the winter for the motorist. Roads here are also plagued by drunk drivers; drive suspiciously, particularly after dark.
There are few significant public-health issues in this region of concern to the traveler, but curiously enough, bubonic plague the real thing, not the DUI "plague" is endemic, and claims a few victims each year most recover with prompt and aggressive medical care. Plague is carried by the small animals of the region, so if you see one in distress, leave it alone and let nature take its course; buzzards are immune to plague, you are not. Drinking untreated water from regional streams is not a good idea owing to Giardia parasites, but tap water is generally not a problem.
One other note: avoid small-town bars here unless you're with someone well known in the bar. Some have clienteles that don't take kindly to strangers.
talk
English, of course, but a considerable number of other languages are spoken in the area. Many residents speak Spanish at home and sometimes at school and work, and dialects of Tewa, Tiwa and Keresan are spoken at the American Indian pueblos of the region, and Jicarilla Apache is spoken in the area of Dulce. Visitors for whom English is a second language may have problems with the indigenous version of English, which is often spoken with a rapid, "machine gun" accent, particularly in some of the rural communities where Spanish is dominant. It doesn't take long to get used to the accent, however. Visitors who speak no English or Spanish at all face some challenges, but a surprising number of residents of Los Alamos and, to a lesser extent, Taos and Santa Fe are fluent in the major European and Asian languages.
One recommendation: If you encounter a place name that appears to be Spanish in origin, it's a good idea to pronounce it as Spanish. A majority of place names in this region are Spanish, some of them with diacriticals to prove it Española, Peñasco, etc., and persistently avoiding Spanish pronunciations will be interpreted by some residents, many of whom speak Spanish at home, as rude. Pronunciation tips in the WikiTravel Spanish phrasebook are useful here; the most common things to watch for are words with "ñ" as in the Española example, double "l" e.g. the very common Gallegos surname, and double "r" e.g. Rio Arriba County, which incidentally is a particularly good place in which to have your Spanish pronunciations in shape.