The Twin Cities (Minn/St. Paul) were one of the first places to pass a TG rights law. Its a pretty progressive and liberal climate. Its also got a pretty vibrant arts deal going on, one of the best rock clubs in the nation (First Avenue). Its very pretty, lots of old homes and nice neighborhoods.
On the other hand, unless you grew up in Point Barrow Alaska or on the North Dakota/Canada border you can't even begin to imagine how friggin cold it can get there in the middle of the winter. And the summers can get very hot and humid. Oh yeah, when they say the state bird is the mosquito they ain't kidding and they are the biggest mosquitoes I've ever seen. The black flys are pretty big too.
However, I would go up in the winter, and when you are out in the woods when its -30 below zero, its very still. And a darn good reason to buy whiskey.
They do have very strong LGBQT groups, and a lot of places to go including the Walker Art Center and the Guthrie Theater is considered one of the best in the world. And you have The Dales - Southdale, Westdale and all those malls, and the Mall of the Americas so shopping is good.
You can go up to the North Woods and canoe for weeks through pristine wilderness (watch out for the Moose, they are very ill tempered) and cross country ski most of the state.