Casual fod:
I agree about the beignets from Cafe Du Monde.(we're about the have some for breakfast!).
Poboys are a must - you can really get them from anywhere but I recommend Mothers or Maspero's. You could get a cup of gumbo (another must) with your poboy and that would be a great meal.
Grab a muffaletta from Central Grocery - across from Cafe Du Monde.
Sit down, nice restaurants:
Commander's Palace - My personal favorite. In the Garden District. Great upscale food made with local ingredients and service is wonderful - it is truly an "experience" when you dine here. If you go for lunch, take advantage of the 25 cents martinis. It is dressy - men wear coats.
Pascal's Manale - more casual than Commander's with more New Orleans style Creole dishes. They are famous for their Barbecued Shrimp. Don't think barbecue in the traditional sense - it's not that at all - it's more like shrimp soaking in a buttery concoction. It's a mess, but to die for.
Jacques-Imos - lots of really good local dishes. Very casual. Go early, it gets very crowded.
Restaurant August - call now for a reservation here. It's in the warehouse district (not too far from the quarter). Dressy and upscale, but one of the best restaurants in the city.
Check out nomenu.com. It's the website of our local food critic.
Weather in March should be beautiful. I'm not sure if it would be warm enough to swim, it might be. Have fun and PM me with specific questions if you have them!