For some really great suggestions, go to City-Data.com > Forums > Minnesota > Minneapolis-St.Paul
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Check the following links below:
For local Scandinavian foods, check out
Ingebretsens. Or maybe the
Bachelor Farmer. There are many more!
One of the premier local restaurants (and featured on Anthony Bourdain's show where he said it was "far and away the best" of the restaurants in mid-America) is
Piccolo. I would also strongly recommend
112 Eatery in downtown Minneapolis. The head chef is a James Beard award-winner. This is the place that good chefs go to eat.

There are many others, but both of these places I'd classify as
"Fresh American Restaurants."
For breweries, I'd check out
Fulton. They have a tap room. Also
Harriet Brewing. They also have a tap room. One of the bigger local brewers is
Summit. But they don't have a tap room.
More information is here.
A quick and easy trip from the Twin Cities to a really beautiful state park is the trip to
"Interstate State Park" on the gorgeous St. Croix River just 25 minutes from downtown Minneapolis or 15 minutes from downtown St. Paul. Check out the pictures from that link.
For non-downtown commercial strips, the answer is fairly easy. Go to
Uptown and/or
Lyndale-Lake in Minneapolis. In St. Paul the easy answer is
Grand Avenue between Dale and Lexington. Another great spot is
50th & France Avenue on the border of Minneapolis and Edina. There are several other spots but those are the premium places to hang out in the Twin Cities other than downtown Minneapolis.