New Orleans is not an American city that learned to be foreign. It is a French, then Spanish, then French again port that was sold to the United States in 1803, and the people who lived in it, Creoles, enslaved Africans, free people of color and later Irish, Germans, Sicilians and Vietnamese, kept doing things their own way. The French Quarter's balconies are Spanish, because the city burned down in 1788 and was rebuilt under Spanish rule. The levees and the pumps are the reason there is a city at all, and in 2005 they failed: by 31 August, 80% of New Orleans was flooded, while the French Quarter and the Garden District, on higher ground, stayed dry.
It is a city to be slow in. Music starts in the afternoon and goes on until two in the morning, lunch is a legitimate activity, and the best sights are small: a tomb, a courtyard, a sandwich counter that has not changed since 1906. Five days is the right length because it lets you do the Quarter without making it the whole trip, ride the streetcar to the Garden District, spend a day in Tremé where jazz and Mardi Gras Indian culture grew, see the two big museums, and still give one full day to the River Road plantations, which are the part of this region's history that most visitors skip and shouldn't.
Each day stays in one part of town. Day one is the French Quarter and the river, day two the Garden District and Uptown, day three Tremé and Frenchmen Street, day four the Warehouse District and City Park, and day five a day trip up the Mississippi to Whitney and Laura plantations. Start on a Thursday: the Whitney Plantation is closed on Tuesdays, the art museum on Mondays and the Backstreet Cultural Museum on Sundays and Mondays, and a Thursday start puts the plantation day on a Monday, the day New Orleans cooks red beans, so you finish with the right dinner. If you want to know what the country around it looks like, there is more in our guide to the United States.