Man I wish that I had seen this earlier.
I’m a native New Orleanean, but no longer live there.
For years though, I had the ultimate luxury of travelling back 2-3 times a year on expense 
I visited most of the famous places in NOLA over the years.
Here are my recommendations for places to eat, in no particular order, both in and out of town:
Mandina’s 3800 Canal Street, New Orleans, LA 70119 (504)482-9179. The quintessential neighborhood New Orleans restaurant. Everyone will know each other. The specialty is trout almandine. You’ll know why after you’ve eaten it.
Superior Seafood 4338 St. Charles Avenue, New Orleans, Louisiana 70115. Very good seafood at fair prices.
Seithers Seafood 279 Hickory Avenue Harahan, LA 70123. Very funky hole in the wall kinda place, excellent food.
Heads & Tails 1827 Dickory Ave, Harahan, LA 70123. Excellent food.
Rum House Caribbean Taqueria 128 Magazine, New Orleans, LA. Taco Tuesday is the thing to do here.
RiverShack Tavern 3449 River Road Jefferson, LA 70121. The epitome of hole in the wall. Good food and beer.
Cooter Browns 509 South Carrollton Avenue New Orleans, LA, 70118. Just drink, don’t eat.
Carmo 527 Julia St, New Orleans, LA 70130. Kinda quirky, but very good
Vincent’s Italian Cuisine 7839 St Charles Ave, New Orleans, LA 70118. Great Italian Food. New Orleans has a large Sicilian community. There are a number of Italian restaurants and Vincent’s’ is my favorite. Vincent is the brother of a friend of ours.
Mosca’s 4137 US-90 West, Westwego, LA 70094. The holy grail. Seriously. It’s a very nondescript place out in Westwego (on the West bank dahlin’). Absolutely sublime. More New Orleans Italian.
Galatoire’s 209 Bourbon St, New Orleans, LA 70130. Bring your best jacket and checkbook. Fiendishly expensive, but worth it. Really old world New Orleans.
Mothers 401 Poydras St, New Orleans, LA 70130. Mother’s is a paradox. Most of the patrons are tourists and get the standard fare, their “world reknown” sandwiches. The sandwiches are good, but I go to Mothers to get the fried chicken. This style of fried chicken was invented by Chef Austin Leslie at “The Chez” so long ago. They cook it to order, so it takes at least 30 minutes. But it’s worth it. Absolutely amazing. I get it with two orders of red beans and rice as sides. Mother’s doesn’t do fries…
Crabby Jacks 428 Jefferson Highway New Orleans, LA. Another hole in the wall. Sublime fried chicken in the same Austin Leslie style.
Sal and Judys 27491 U.S. Hwy 190, Lacombe, LA 70445. This is over on the North Shore of Lake Pontchartrain. Worth the trip, again more New Orleans Italian food.
Salvo’s 7742 LA-23, Belle Chasse, LA 70037. Local seafood place again on the West Bank in Belle Chasse. It’s where I go to get seafood when I’m in New Orleans. Very unpretentious local joint, but the seafood is fresh off the docks. Oysters, crawfish, shrimp, etc. They even serve dinner now.
Several (most) of the places mentioned earlier are tourist traps. They are ok, but nothing to crow about. The places listed above are where the locals eat and drink.
New Orleans isn’t the safest place in the world nowadays. Keep your wits about you and travel in groups, particularly after dark.
Bon appetite!