If you asked me the epitome of New Orleans in one dish, it's gumbo- hands down, no contest. The answer to "where to get the best" is always "my mom's house"; it intertwines Cajun, Creole, indigenous native, African, Spanish and French. And the biggest basic you need to know that a recipe can't teach: the roux, your base of flour and oil, MUST be a dark brown, like 60% + bittersweet dark chocolate color. None of this blonde roux business- make it mahogany.
Traditionally it's served with a scoop of rice, but some folks do potato salad. Don't argue with them, just nod and take your bowl. Chicken and sausage is the most common in a restaurant, but seafood is where it's at- always opt for the seafood version. Okra is the vegetable that thickens it, but is often left out sadly. File is the ground up leaves of the sassafras plant added at the end to thicken it sometimes. Both are credited as possibly being where the word gumbo comes from (the words for okra, the African, for file, the Choctaw)!
Gumbo Z'Herbes is a traditional Good Friday/Easter gumbo that excludes meat, in observance of Lent. Traditionally, it uses 9 types of greens in some sort of symbolism to the holiday and folks used to use an herb called pepper grass that they'd pick right out on the roads. The Creoles use mostly seafood in their gumbo, while the Cajuns throw in anything that swims, crawls or flies to feed big families and neighbors. The rules are not steadfast of WHAT goes into gumbo, but love and patience are the most important part of any gumbo.