For the Juicy (or Jucy depending on the location) Lucy, Matt's Bar is the local standard for it but it sucks. Shitty meat, shitty bun and velveeta cheese. For a better example, go to the Nook in St. Paul.
For good food, check out Saffron, 112 Eatery, Travail Kitchen & Amusements, Piccolo, Bar La Grassa, Restaurant Alma, Heidi's, and Tilia. The first four I listed are probably my favorite restaurants in the Cities. There's a huge SE Asian population here also, so there's great Vietnamese food. Saigon, Pho 79 and Quang are my top 3 in the category. If you want Somali food (the Cities has the largest population in the US), Hamdi or Safari are your two best bets. Hamdi is more authentic but you'll be the only non-Somalis there, while Safari is quite accessible.
The Minnesota Institute of Arts (M.I.A.) is a really good museum and free. The sculpture garden is part of the Walker Museum which specializes in modern art and often has very cool exhibits going on.
Visit the lakes, you can either walk or rent a bike at these kiosks spread all throughout the city for cheap. Great people watching and they're incredibly nice to lounge around in the summer time.
Drink as much Surly as you can. It's a great local brewery that just got some landmark legislation passed that's very pro-beer (if you care about such things).
Minneapolis also has one of the highest theaters per capita in the nation but even if you don't see a show, visit the Guthrie. It's right on the riverfront so it's a good place to walk around regardless. I don't know dick about architecture but it won several awards when it opened so apparently it's quite renowned.
As for night life, your age and interests would be helpful. I'm sure your friends have their favorite haunts, but Uptown is a good place to go; Barbette's is my favorite place there. Downtown has a few nice spots as well.
There's a thread on the Twin Cities that's not that old you can search for also for more recommendations.
