It’s a touristy thing to do, but I enjoy hanging out at the Café du Monde, sipping cafe au lait and muchning beignets while I watch the people pass by, listen to the music, and feel the breeze off of the Mississippi.
There’s another coffee house across the street from Café du Monde. Not so many tourists there, and on a nice day it’s nice to sit by the elevated, open windows and people-watch. I can’t remember the name of it right now, but I’m sure someone will tell me.
The French Market is fun. Try the gator-onna-stick! Mmmm.
Marie Laveau’s House of Voodoo is worth a look, but only because she’s so famous. The Voodoo Museum (on Dumaine?) is better. Marie Laveau’s tomb is just outside of the French Quarter (I forget the name of the cemetary). Make three Xs, knock three times, leave an offering of tobacco, alcohol or money, and make a wish! (Didn’t work for me, though. I’m still single!)
Also on Dumaine (I think) at Bourbon is The Clover Grill. I like a burger with a side of hash browns. And the staff are fabulous! 
There’s a funky little Thai restaurant on Esplenade, near the French Market. Saim House? Something like that. Go upstairs and sit on the floor. Good food.
Checkpoint Charlie’s is a rockin’ little bar with live music and very good burgers. You can also play video games and do your laundry there.
The R Bar is good, but I like Checkpoint Charlie’s better. There’s also the Saturn Bar. It’s a real dive, and has the weidest collection of stuff in it.
I think Jean Lafitte’s Blacksmith Shop is the oldest building in The Crescent City. It’s a bar now. Worth it just to say you’ve been there.
Just walk around the Quarter. The architecture is wonderful.
Ride the streetcar into the Garden District to see a bunch of nice old houses.
Go to the Audubon Park and see the nutria!
Walk along the river, or take the ferry over to Algiers. There’s not much in Algiers, but I had fun walking around it.
Visit the cemetaries.
I’d avoid Royal St., but just because it’s too chic-chic-buy-this-stuff-y.
I love The Big Easy. It’s my favourite city! I thought about moving there once, but my friends moved away. I could live there if I had friends there, but otherwise the summer heat is opressive.