I go a few times a year so maybe I can help, and we tossed around the idea of going on new years but figured finding a place to stay would be pretty difficult. If you can I would recommend staying in the french quarter if you can find a room. That way it's really no more than a $10 cab ride anywhere you would want to go.
Bourbon Street is where the big party is going to be but I'm assuming that it will be slammed. My favorite places on Bourbon are the Cat's Meow (awesome karaoke bar and one of the few place you can get on a balcony without having a balcony room), Pat O'Briens (gotta stop in for a hurricane), Razoos (pretty cluby but still fun), and my new favorite is called the Dungeon (heavy metal bar, go to the back room)
The other cool place to party is down on Frenchman street, there is a lot of bars that mostly play real live jazz and has a more local feel.
The food there is some of the best that I have ever had. Go to Central Grocery, 923 Decatur St. and get a muffuletta (the best in the world IMHO). The Gumbo Shop and Coop's Place both have great food and it's very reasonably priced. Crescent City Brewhouse is great but pretty expensive. I can honestly say that I don't think I have ever had a "bad" meal in New Orleans, just keep it simple and stay away from chain restaurants like Bubba Gump and TGI Friday's and the like.
There are carriage rides all over by Jackson Square. We have done a few and it's always really interesting, there is a lot of history in New Orleans, also Royal Street has ton's of little art shops and antique stores if your into that, good way to kill time and burn off some alcohol.
I don't have a lot of info on the Superdome other than it's downtown and pretty close to the french quarter so getting there shouldn't be a problem.
Have fun, it's my favorite city to party in!!!
Hope that helps