I work in a property tax office and have for 15 years, so this is something I know a little about.
You can use 1% as a rule of thumb when calculating what your taxes will be, at least in Dona Ana County. So a 100K house will have an annual bill of 1K. This contrasts to El Paso, TX a little south, where you can expect to pay about 4%. Of couse Texas doesn't have a state income tax so they have to get their money somewhere. Even so, with our income tax and property tax combined it's less than what you would pay in Texas.
There is also a limit (3%) to the amount of increase in value you can see from year to year for residential properties, in addition to head of household, veteran and senior exemptions or value freezes. This, while values are increasing at about 10% a year.