I'll add an enthusiastic vote for a visit to A Gallery, always visit it wishing I had a few thousand dollars to invest in a classic photograph; and the above-ground cemeteries absolutely, and Mothers for breakfast downtown, just outside the Quarter. Look for the line of people waiting to get in - it is part of the experience.
You must also have a beignet plate at the original Cafe Du Monde on Decatur - do NOT wear black there, and check out the clubs on Frenchman Street, especially Snug Harbor, a well known jazz club/restaurant. And the last time I was down there I really enjoyed an upscale dinner in a little place called Bistro at Masion De Ville in the Quarter. Great selection of wines, wonderful seafood.
Great for street photography, informal portraits of wonderful characters and street performers, and authentic architecture.