Try Flagstaff. They get all four seasons, and they recently got dumped on by a nice big snowstorm. You can ski, and do all kinds of neat outdoor wintry things there.
Like the first answerer, Flagstaff is one of the towns in Arizona that snow. Although it is quite pricey. I have a friend who lives there, and I visit him a lot. His rent is about $500, he splits a 2 bedroom with a roommate. Since Northern Arizona University is located there; it is a college town. It just had it s first big blizzard come through about a week ago, but it s quite clear of snow now. It has quite a few of trendy clubs/restaurants/cafes and shops downtown where all the partying goes on. It s a nice town though, and has a population of about 60,000 people. I am moving there after I graduate this year, and am planning to attend NAU.EDIT: It all depends on the quality of the apartment also, but hey, if it works for you :)
Flagstaff snows I live in Scottsdale and its snowed here in the past but in the mountains it usually snows.
Flagstaff is the snow city in Arizona. Sedona also but very expensive.