Po boys - parkway tavern or verti marte
Restaurants - Maypop, Toups, Bacchanal, compare Lapin, ralphs on the park. Camellia grill - its a diner but it has some charm, good hangover cure.
Bars - r bar, black penny, bayou beer garden (usually has crawfish on the weekends when in season) Mimis in the marginy - excellent food too but it wouldnt seem like it. And Mollys at the market. These are just a couple of the places I hang out at.
These are just some ideas I havent to seen mentioned yet and some places I think you will find more locals than tourists. There are plenty of old school New Orleans restaurants that are easy to google if you want to check those out. Commanders, Brennans, Galatoies to name a couple. And google whats going on the week your coming, theres always festivals and whatnot, specially near the quarter in Armstrong park. To check out a different side of New Orleans take the fairy to the West Bank, couple of bars in the area it drops you off and it gets you out of the chaos that can be the French quarter.