The French Quarter is great if you have never been. Walk around and absorb it all.
If you like great live music, I would suggest you head to the Marigny. Its a neighborhood that boarders the French Quarter and has some of the best live "New Orleans" music in the city.
There are so many places to eat and I am sure you will get a ton or rec's, but if you like oysters you have to go to Drago's and get the char grilled oysters. Acme is fine, but there is always a crowd and that's where all the tourists end up. Any of John Besh's places are great to eat. It really depends on what you are looking for. One piece of advice though, don't eat any "cajun" food. That's for the tourists since there are almost no cajun's in New Orleans and it's typically not good. If you have never eaten a Muffaletta, go to Central Grocery and get one.