Good recommendations so far but, as a New Orleanian, I'd like to add a few:
New Orleans Photo Alliance Gallery:
http://neworleansphotoalliance.org/grants/CJL_Award/index.php. The Four by Five exhibit will be up. If you can't make it during regular hours, PM me and I'll meet you there.
Ogden Museum of Southern Art:
http://www.ogdenmuseum.org/.
Across the street are the WWII Museum and the CAC (Contemporary Art Center)
Julia St. has lots of art galleries. There's a wonderful show of Gordon Parks unpubllshed work at Arthur Roger gallery (
http://arthurrogergallery.com/exhibition/gordon-parks/)
In the French Quarter is A Gallery for Fine Photography (
http://www.agallery.com/) Your eyes will pop out of your head at the great photos here.
The Historic New Orleans Collection (
http://www.hnoc.org/) Wonderful photo exhibit by Richard Sexton.
On Jackson Square, just down the block from Cafe du Monde, is PhotoWorks Gallery. (
http://www.photoworksneworleans.com/)
The LA State Museum is worth a look. (
http://www.crt.state.la.us/louisiana-state-museum/index)
Just off of Frenchmen St. is Scott Edward's Gallery (
http://scottedwardsgallery.com/). Great selection of contemporary photography.
The New Orleans Museum of Art is worth a visit and it sits in City Park which is a beautiful place to relax. There's a Morning Call Coffee Shop in the park as well.
That's just scratching the surface. As for food and drink just stay off of Bourbon St. and you'll be fine unless you go to Galitoire's or Lafitte's Blacksmith Shop. Definitely check out Preservation Hall and any of the venues on Frenchmen. Chickie Wah Wah on Canal St. is awesome:
http://www.chickiewahwah.com/; non-smoking, good food, great sound system and good selection of local beers.
Have fun!