Minneapolis is considered one of the best cities in the country for Vietnamese food. Quang on "Eat Street" (a section of Nicollet Ave. just south of downtown) seems to be a perennial favorite.
I'd also suggest trying East African food while you're here - Minneapolis has the largest concentration of Somali immigrants / refugees in the country, and they've opened up many restaurants. For a soft landing you can go to Safari Restaurant, which may also have the world's best basmati rice (though eating your suqqar or whatever else you order with pasta may be more authentic).
I'd recommend Al's Breakfast for a great local breakfast scene (and wonderful food), and possibly Hell's Kitchen too (another great breakfast place, though it serves lunch too).
413 14th Ave SE, Minneapolis, MN 55414
89 S 10th St, Minneapolis, MN 55403
In terms of fine dining, here are my suggestions:
If you really want a taste of the Twin Cities, I'd highly recommend Heartland in St. Paul. Everything on the menu is sourced from the Midwest with an emphasis on locally produced items. Chef Lenny Russo crafts a nightly changing menu and the wine list is extremely well-thought out. There is also a wine bar adjacent to the dining room where you can start and/or end your evening. It's the definition of Mn Nice.
In Minneapolis, the standard-bearer is La Belle Vie. It always is usually named best Mn restaurant by most critics in this town. They have a sister restaurant named Solera that features a wide selection of tapas which may be ideal for the type of day you discuss.
The only other thing I'd mention is be sure to hit the Town Talk diner for adult malts. It's a throwback to the old day with modern touches on food and drinks.
Minneapolis doesn't have the wealth of choices that NYC has, but we've got some really nice places that can hold their own.
We've got some great hole-in-the-wall Southeast Asian places, mostly in St. Paul. (Have you ever had Hmong food? There's a large indoor Hmong market in St. Paul.) And we've got a large Somali community, so we have quite a few of Somali restaurants, including Safari (my current favorite at the Midtown Global Market).
The Midtown Global Market is a fun place to try several types of food - Mexican, Somali, Scandinavian, Middle Eastern, Jamaican, and more.
For Farmers Markets, go to either Minneapolis' Mill City Market (local/organic) or the Saint Paul Farmers Market (all local/regional).
For more ideas, see The Dairy Queen's wonderful post for great places in MSP:
I'll keep my fingers crossed for good weather for your visit - going to Sea Salt/Minnehaha Park or Tin Fish/Lake Calhoun is really a must!
Is there anything that you are particularly interested in? Cuisines? There are lots of opinions here, so it'll be helpful to narrow things down.
If it's a nice day, nothing beats a meal at either the Tin Fish (by Lake Calhoun in Uptown) or at Sea Salt (by Minnehaha Falls and close to the Mississippi River). Both are very casual.
Sea Salt Eatery
4801 Minnehaha Ave, Minneapolis, MN 55417
The Tin Fish
3000 E Calhoun Pkwy, Minneapolis, MN 55408