Here are a few things that I would recommend:
Food:
Central Grocery - Muffulettas 923 Decatur St.
Cafe Du Monde - beignets/chicory coffee just off Jackson Square - don't wear black, because the icing sugar will be everywhere!
Cafe Amelie - 912 Royal Street - cheap beer, turducken salad
Mothers Restaruant 401 Poydras Street at Tchoupitoulas - Famous Ferdi $11.75 (roast beef) - walk straight down Poydras or go back to Canal to Tchoupitoulas (7am-10pm)
Clover Grill late night eats - 900 Bourbon St. - it is an interesting place
A bit of gay atmosphere.
Parasol's 2533 Constance St. (11am-9pm)- my opinion the best po' boys and get them with a cold Heineken
Huck Finn's Cafe - 135 Decatur Street (Straight down Iberville towards Miss. at Decatur on the r) - get the Jambalaya wrap and Andigator beer
Acme Oyster House - a must!!!
Lots of fancy restaraunts too - reserve before you leave as you are going at the busiest time of the year - my favorites are Arnaud's (Creole) 813 Bieville St. (you get to eat bread with you hands here) and Galatoire's (French) 209 Bourbon St. suit jackets are mandatory for men for dinner.
Drinks:
Arnaud’s French 75 opens at 5:30 (Chris Hannah is the bartender) 813 Rue Bienville, huge drink menu but I love the Sazarac.
Lafitte's Blacksmith Shop 941 Bourbon St - a bit sketchy - get something in a bottle
Pat O'Brien's Bar 718 St. Peter Street - for Hurricane's - if you return the glass you get some money back so don't let the waitress take your cups!!! Also dueling pianos.
Napoleon House - 500 Chartres St. - get Blackened Voodoo Beer or a Pimm's cup
A few fun drinks to try - Brandy milk punch, Ramos Slow Gin Fizz, Sazarac, Purple Haze beer (any Abita beer is good in my opinion, wish we could get it here), Mint Julep. On Bourbon St. they have a frozen Black Russian that is awesome - just look for the frozen drink machines.
To do:
Research which parade routes you want to see for Mardi Gras - I would do the Zulu and Rex they are the biggest and start it off. Zulu starts at 8:30 am near the Central Business District, Rex is uptown at 11 am. I would also look for the Mardi Gras Indians - they are my favorite piece of history and each costume takes a year to create by hand. They should be in the Central Business District in the early afternoon. Mardi Gras is the only acceptable time for tourists to wear beads, otherwise it is not done. Also for Mardi Gras you can dress up as anything you like for the parades. Be very careful of pick pocketers though... Also after dark don't venture out into Armstrong Park or past the Market, it is just not too safe. Also flashing for beads is something only tourists do - just FYI.
Steamboat Nachez 11am boarding leave at 11:30 - 2:30 right on the Mississippi - they also have a jazz night cruise - it broke down when we went - LOL
Blaine Kern's Mardi Gras World (9:30 - 4:30) 1380 Port of New Orleans Place NOT on OLD ALGIERS!!!! - take Riverfront Street car Bienville station to Julia station or walk
They have Haunted History Tours - I would do the vampire one if I had to choose but they are all pretty silly
Garden District walking tour - at Jax Brewery you will find a tourist centre that can hook you up or the gazebo outside of Jax's by the river. You can also get a guide book and do your own walking tours. Make sure you go to the Garden District book store - they have great local authors and many of the books are signed.
Street Cars - St. Charles Avenue to see the estate homes, a great piece of history. If you get off at Tulane University (I think this is the stop) it is a great walk around Audubon Park.
Plantation day tours. Great history and you have your pick of plantations.
Music:
You will have to research what you like but Preservation Hall is great. Standing room only, I think they open at 10 pm and it is $12.
FYI there will be a ton of street performers - if you watch then you pay.
Blues - so many places to hear music.
Hope this helps - it is all from my running info on NOLA.