Don't forget Mid City!
I always have friends ask where to go in the French Quarter when they visit, they almost always miss the great eats in the Mid City area. Places like Juevos, Juan's Flying Burrito, Mandina's, Ye Olde College Inn (Get their fried green tomato PoBoy) and many others are favorites of locals. Hit a few of these places and check out City Park, it's worth it.
And try the bars on Royal instead of Bourbon...