Look around and you'll see a couple of posts on the subject but basically
For climbing, depends on what you are looking for and at what grades. what you will find is a ton of it close by, between 5 minutes to 45 minutes depending on where n town you are and where you want to go. Climbing is nearly year round, if you are willing to adjust you altitude. In the winter there will be a few storms that drop significant snow at the lower altitudes, and Flagstaff can have a hell of. Lot of snow ( or none at all), but temperatures above 90F are basically unheard of, and outside of June, summer temps tend to be in the 70's maybe low 80's...
For trad climbing, it's pretty damn good. Paradise Forks is pure good fun, but if you are a 5.12 trad leader, you'll out climb it pretty fast. If younoutclimb the Forks then there is the Waterfall. Some of the best climbing in such a condensed area then I have ever seen, but it really starts at 5.11 ( a couple of good 5.10's) and goes into the .13's with the bulk at 5.11/.12). Sedona: can be chossy in areas and can be pure bliss. There are other areas that are going up.
For Bouldering: I'm not much of a boulderer, but the area has tons of it at all grades
For sports climbing: very strong in the moderate to fairly high grade, but if you are looking for a ton above mid 5.13 you'll be disappointed. There is some but not a lot. In the 5.8 to 5.12 as I said, lots with new areas regularly
Ice: better look elsewhere since the are a couple but it's pretty bleak
For people: you'll find the local community to be pretty laid back and fairly accepting of new people....Either go to the gym, or ask about the GNARbarian meetings ( ok, not a meeting but a reason to go to a brewery, pizza joint, and tells stories, make plans, BS, make friends, get drunk)
Cost of living: the one downside. if our are one of us who works for either WL Gore, the Hospital complex, or NAU, it's not to bad....for everyone else, it is pretty pricey, especially housing. But lots of climbers make it happen.
I moved here 7 years ago from the Bay Area, and for me, it was an awesome move.its not for everyone, but for a lot, it's a pretty good place.