Some are touristy, but it's New Orleans, so most things are. Here's what I like:
Lunch at Mother's. Its a tradition and the food is excellent. Missing Mother's is not an option.
Breakfast at Lucy's Retired Surfer Bar & Restaurant. The warehouse district is getting more popular these days.
Steak Dinner at Mister Bs or Dickie Brennan's or Delmonicos or ...who am I kidding? Anywhere! The Duck at NOLA (Emerill's place in the quarter). And of course, cajun and creole food just about anywhere.
Walking the Riverfront Mall. Some good food and typical mall stuff, but it's something to break up a day and get out of the heat and come home with a souvenir.
Some swamp tours pick up in the city and drive you out, so you won't have to rent a car to do a swamp tour.
Midnight to 2am at the Funky Pirate with Big Al Carson. Best blues in the quarter and its usually pretty crazy by then.
Hurricanes at Pat O's. Great for people watching too.
Catching a nice local act at Margaritaville.
D-Day museum. (worth a trip, really)
Frenchmen Street for music.
Take a cab uptown to Jacques-Imo's if you want awesome creole food. Meaning, the best in the city by far.