If you get there in time, go to Friday lunch at Galatoire's. It is very much a locals' scene with the Nawlins' socialites all out meeting and greeting. An interesting scene!
Other personal favorites would include Acme Oyster House, Tujagues, Court of Two Sisters for Sunday brunch, Lucky Changs (very unique servers) and Cafe du Monde for coffee and beignets in the wee hours.