Looking forward to these answers. I remember being a 15 year old, walking down Bourbon Street with my parents. It was quite the eye opener.
There are nice carriage tours of the French Quarter.
There are river boat cruises which will take you up river to plantations.
You can take the St Charles trolley out to Audubon Park and Zoo.
There is an Aquarium on the river front, just outside the quarter.
Cemetery tours in New Orleans are exceptional, as the cemeteries and their practices are very different than the rest of the country.
Coffee and beignets at Café du Monde.
Walk the French Market
Muffaletttas at Central Grocery, near the French Market.
Oysters at Acme Oyster House, maybe a tray of crawfish and an oyster or shrimp po-boy.
Bon Ton Café, just outside the quarter, in the business district is a great place for classic New Orleans cuisine.
Pascale's Manale off St Charles on Napoleon invented New Orleans style barbeque shrimp. That and their crusty bread makes for an awesome meal.
Tujaque's, near the French Market is the oldest spot in the quarter and a great place for lunch. Excellent gumbo!
Court of Two Sisters is a great place for a jazz brunch - incredible buffet.
I will say, New Orleans is much better suited for adults than children. Would suggest you stay off Bourbon. Otherwise, have a great time.
psudad You did a great job with your list of suggestions for a family with teens. For another great meal, you can try Parkway Bakery and Tavern. It recently won the best shrimp poboy in Louisiana contest and I fully agree.