Oh where to begin, NOLA is my second favorite place in America behind Monterey, CA.
Don't get there nearly enough but here is my advice:
Be aware. At ALL times. It is a wonderful and glorious city but it is shady as hell too. Crime is very real there.
The WWII Museum is fantastic. Been there last two trips.
If you are an early riser, watching the sun rise over the St. Louis Cathedral is great and it's a hop from there to the Cafe du Monde for the Chicory Coffee and Beignets.
From there you can walk back in to the Fr. Quarter and see the sights. If you get far enough down Bourbon St, it switches teams, if you know what I mean. No big deal but be aware it may go from girls offering to buy you drinks to boys offering to buy you drinks.
I Love the Garden district. You can take a trolley right down St Charles Ave and visit Toulane U. and see some gorgeous houses.
St. Louis Cemetery is pretty awesome too.
Restaurants: I have been to a few awesome places (in no particular order):
St. James Cheese Co.
Antoine's Restaurant I literally wept when I learned they lost 10,000 (!!!) bottles of wine to the Katrina flooding... Great food though!!
Cafe du Monde Chicory Coffee and Beignets nom nom nom
Clancy's Restaurant Great Place to eat after the Auduban visit.
Commanders Palace CP is AWESOME out in Garden District Food devine. Only down here was the staff thought that my (then) 15 y/o step daughter was my girlfriend. yuck.
Southern Candymakers
If you have a car and want some adventure you NEED to do an airboat trip once in your life.
http://www.airboattours.com/ One of their Captains was on the show 'Wife Swap'. It was epic. All that aside, this tour is AMAZING. you go from tidewater into a private freshwater reserve filled with gators and all manor of flora and fauna. Highlights are their 13 foot gator that they feed for you and bring up next to the boat.
Have a GREAT Time!!