Let me also chime in here being a "recent" transplant from Chicago.
Unlike Chicago, there is no local Minnesotan food that is really iconic (like the hotdog, italian beef, giordanos pizza, etc.). Yes there is lutdefisk, hot dish (think casserole), and other swedish/norwegian "delicacies", but they arn't something I would spend any money on.
However, that does not mean that there arn't any places to get really good food. People in Minneapolis regulary go out to eat (because its so darn cold, eating makes us feel better).
At least that is what I like to spend my money/time doing
This is why I asked the leading question in a different post with regards to what kind of food you like, because you can find a restaurant that will make it here.
The Happy Gnome, Muddy Pig, and my personal favorite The Bulldog (located in St. Paul) all serve awesome beers (their beer menus are books to page through) and server good gourmet burgers; the Bulldog has AMAZING tater tots (e.g., wasabi tots, truffle tots, caijun tots).
Outside of that type of food, Chino Latino (Mpls) is a fusion restaurant of Chinese and Mexican food, and is a fun place to go to.
I am also a fan of Barbette (Mpls), it is a french bistro (not 5-star but 5-star food), their eggs benedict and bloody marys are fantastic!
If you want to spend the money, Murrays is my top recommendation. Expect to spend a good chunck, but Murrays is a classic americana steakhouse/restaurant where they serve you meals on china, think going to grandma's for sunday dinner but getting 5-star food (sorry grandma), it is truely an experience worth doing at least once.
There is sushi, vietamese, chinese, ethiopian, italian, french, fusion, fish, steak, burgers, and general "american" restaurants to pick from.
Of course there are tons of chains (applebees, etc.), but there are also so many more options for good food that won't necessarily break the bank. It really depends on what you are in the mood for, and which city you want to hang out in.
If you want more info, or if something catches your eye let me know. The wife and I go out regularly, and while we haven't eaten our way across the cities, we have been around the block once or twice.
-Ryan