Ok. I'm belligerently drunk and am ready to write a review, so bear with me. Born in raised in Algiers, LA. That's across the river from N.O. (a whopping 4 miles by highway, or 500 yards by ferry, which still operates).
First off, the food:
I've heard a couple of recomendations for the Red Fish Grill. Skip it. It's on Bourbon and if you are in the Quarter it's convenient, but as a local, I'm not a fan. After BP I went their for oysters. Bastards served me Atlantic oysters and called them Gulf oysters. A criminal offense in my opinion. So, my recomendations:
Best po-boys:
(1) Domilese's. It's in the middle of Uptown, so those staying the Quarter will need to take the St. Charles street car or take a cab. The best po boy hands down. That being said, a limited menu. Basically fried everything + your standard roast beef. But if that's what you want, go there.
http://www.yelp.com/biz/domilise-sandwich-shop-and-bar-new-orleans#queryo%20boy%20sandwich
(2) Mahoney's
On Magazine St. So again, if your in the Quarter it's a bit of a commute. This is where you should go if you want a more ecclectic menu. My recomendation, the Oyster-cheddar-bacon-romelaude po boy. If you don't like that, have a sex change cause you are not a man.
http://www.yelp.com/biz/mahonys-po-boy-shop-new-orleans#queryo%20boy%20sandwich
Best high class cajun:
Cochon. They have my hands down favorite dish in the city. Rabbit and dumplings. Also, cocktails are wonderful. Make a reservation as the place gets packed. It's in the Warehouse District, so either a quick cab ride or a decent walk from the Quarter.
http://www.yelp.com/biz/cochon-new-orleans
Best classic Creole/French:
Commander's Palace
Located again in Uptown, but worth it. If you want the white linen traditional french experience, this is where you should go. A bit dressy, so be prepared. Also, pricey. If you want to save a penny, go for the Sat-Sun jazz brunch. $30 for three courses (do not skip the bourbon bread pudding suffle) + a little extra for booze. If you put down all three courses, then you can stand with the locals.
http://www.yelp.com/biz/commanders-palace-restaurant-new-orleans
Best overall restaurant:
Lilette. A small French restaurant on Magazine. A reletively new place, but simple put, the best restaurant in the city. Dressy, and reservations are a must. But you won't find better food.
http://www.liletterestaurant.com/
Best place to drink, have a swim, and stare at tits:
The Country Club. A clothing optional pool in the Bywater. Now, many a tourist skip the Bywater as it borders up against the 9th Ward and people deem it "sketchy." But, you will find no locals about. A word of warning though, a somewhat gay area, so you'll see a bit of dick at the pool. If that weirds you out, avoid. Nonetheless, a good place for a good weather cocktail.
Best place to feel like a local:
Bacchanal. An outdoor, live jazz venue in the Bywater. Made a bit popular from the HBO series Treme, but still mostly locals. Good food, no tourists, get hammered and take a cab home.
http://www.yelp.com/biz/bacchanal-wine-new-orleans
Best Oysters:
Casamento's. Uptown, and only open from Thur-Sat.
Beignets:
Cafe Du Monde. If you're in the Quarter, this will be walking distance for you. Don't try to go for breakfast, it will be packed. Instead (as it is open 24-7), stop buy after a night of drinking before crashing.
In regards to a few other posts:
-Skip Acme's. Perfectly fine, but generic and a huge tourist trap.
-Skip Mothers. Overpriced and nothing special.