bad stuff first. Do not go into cemeteries alone. You stand a very good chance of getting mugged. The high walls keep the crime from being seen by anyone.
If someone wants to clean your shoes, or guess where your shoes came from, or wash your car windows, your guard should be up - they may be trying to distract you so they can rob you.
STAY OUT OF THE PROJECTS. Think Chicago's South Side, only worse.
Now the good stuff.
Cafe du Mond in the french quarter is THE place to go for breakfast. Order bengets and French Market coffee.
Tujagues is a good expensive restaurant. Great creole food.
Ralph & Kacoo's is a good less expensive seafood restaurant. The hush puppies ROCK there. You can't miss 'em either - the buildings are a rather startling pink.
Antoine's is a good french restaurant.
Whatever you do, find a place that bakes their own bread (preferably on the water) and sells oyster po'boys. You have GOT to try these things!
As far as eating, just about anywhere in New Orleans is gonna be good to eat at - the best food in the world is there and if someone's food isn't good, they're laughed out of town
The French Quarter is kinda interesting, but honestly it's gone way downhill in the last couple'a decades. The sex industry has taken over, which is a shame. As far as I'm concerned, you can get sex toys online, and you don't need to buy them in the formerly quaint old shops of the Quarter. It's changed the atmosphere quite a bit, and not for the better - you can get porn anywhere, but it's hard to find good artistic communities - too bad this one's rapidly disappearing.
all that aside, however, the street musicians in the Quarter are almost to a man better than the stage musicians anywhere else. You'll see guys with about $4 to their name playing jazz better than anything you've ever heard back home.