Having lived in New Orleans (aka The Big Easy aka The Crescent City aka The Birthplace of Jazz aka The Home of The Go Cup aka The City That Care Forgot, etc.) from 1978 to 1983 I had the opportunity to check out many of the sites. I haven't been back since the early 90's so some of these places may not be around any more.
There are several touristy type places worth checking out and most can be found in the French Quarter. Some of my favorite FQ restaurants were K-Paul's Kitchen (owned and run by famous Cajun chef Paul Prudhomme), Felix's Oyster House (try the gumbo and the Oyster's Bienville), Brennan's (brunch at Brennan's is a tad expensive but a must for a first time tourist - start by having a Milk Punch or Ramos Gin Fizz while hanging out in the courtyard). After eating, head over to Pat O'Brian's for a Hurricane Punch. If it's late, late at night, try The Dungeon (it's where the locals who work the FQ clubs go after they get off).
IMO, the best clubs and restaurants are found in the Uptown/Garden District/University area (easily reachable by taking a streetcar from the French Quarter). My favorite was always Tyler's Beer Garden on Magazine Street. I'm sure the prices have gone up but when I used to hang out there raw oysters were 10 cents each, crabs were 50 cents, a plate of crawfish or BBQ shrimp was $2 and a pitcher of Dixie beer was $2.50. Great local jazz musicians at night also.
Tipitina's on Napoleon Ave. is a great place for live music. The Camillia Grill located at the riverbend (where St. Charles turns into South Carrollton) is a must for breakfast. If you really want to get jiggy with the locals try Nick's Bar on Tulane Ave. (caution: not a great part of town).
Mardi Gras is not the best time to be in N.O. (in fact it's probably the worst time) but if you like to eat, drink and listen to great music I'm sure you'll have a wonderful time.