( Updates in 2018-02: Add these to your list: Willa Jean, Mammoth Espresso, Doris Metropolitan, Lattitude 29, Spitfire Coffee, Paloma Cafe )
Since we're in the CBD, my favorite nearby places Cochon (but I really like butcher, it's less formal), Peche, Compere Lapin, August ($$$), Willa Jean, Juan's Flying Burritto (CBD location), Carmo, Luke on St. Charles, (great happy hour)...
Nearby for coffee: Revelator Coffee
Nearby for wine: Keife & Co, W.I.N.O
Nearby bar for hangout: Lucy's Retired Surfer, Vic's Kangaroo
Stuff I'm going to check out this trip in the area that has opened recently: Bakery Bar, Espiritu,
Magazine Street - Starting from Calliope, and running uptown, Magazine is a funky shopping district with lots of interesting independent stores. Check out Juan's Flying Burrito for awesome creole Mexican food. Cheap, filling, and good quality. (And loud music.) Lilette is expensive fine dining near Louisiana Avenue.
If you're uptown, check out Oak Street. Zotz is a cool coffee shop. For dinner, you can try Jacquesimo. It's a little pricy, but decadent new orleans style food. If you want that same type of food for cheap, Crabby Jack's on Jefferson Highway is run by the same guy. My favorite there is Blackened Gulf fish with crabmeat remoulade sauce. Their Duck & Andouille gumbo is pretty damn good, too. Freret Street between Jefferson and Napoleon has become a culinary wunderbar. Amazing because when I lived there it was dangerous and all boarded up. Breakfast at bearcat cafe is excellent. High hat is New Orleans style. Ancora Pizza is well regarded.
Museums: Ogden (regional folk and outsider art), Museum of Modern Art, Contemporary Arts Center. The D-Day (WWII) museum is supposedly really good. There's also a civil war museum.
Ride the St. Charles Streetcar line uptown to Audubon park, or go on the Canal Streetcar line to Mid-City for City park and the botanical gardens. If you're in Mid-City go to Angelo Brocado's Italian bakery for cappuccino, gelato, and cookies. It's at Carrollton and Canal Street.
Vietnamese food in NOLA is some of the best you can find in the country. My favorite is Nine Roses. It's on the west bank (of the Mississippi) and a little hard to find, but we have big family style meals with between 5 to 20 people. It's an amazing feast, and ends up being about $25 per person.
For nice quarter restaurants, I suggest Bayona, NOLA (Emeril's place), Pelican Club, Mr. B's, or Palace Cafe. They're all fairly expensive. Tujaque's is old school New Orleans. Have lunch at Napoleon house. Definitely go to K-Paul's (Paul Prudhomme's cajun restaurant). Find Cochon Butcher (butcher is the cafe style, Cochon is fine dining style) in the CBD (other side of Canal) for lunch. Any of Donald Link's restaurants are great. My current favorite restaurant in New Orleans is Restaurant August.
In the Treme (on the edge of the french quarter) check out Lil Dizzies for great New Orleans fare.
Check out Frenchmen Street. You can go to Snug harbor for burgers and a Jazz show. But, most people go to port of call on Esplanade for burgers. There are several good clubs on frenchmen St.: Maison, DBA, Yuki, etc.
The Bywater has several excellent places. Maurepas Cafe (update: CLOSED), Bacchanal on Poland Ave. Satsuma's for breakfast or lunch, Cake Cafe for breakfast /lunch.
You can get good coffee at Cafe du Monde, but also Envie at Barracks and Decatur. For excellent espresso drinks, I like Velvet across from Whole Foods on Magazine. There's now a HiVolt uptown also on Magazine by Whole Foods.
There's the original Hi-Volt on Sophie Wright place(near Magazine in Garden District near that Juan's), Mammoth Espresso and Spitfire in the CBD & Quarter. Hi-Volt also great breakfast / brunch and baked goods, but they're much better at Sophie Wright location.
Try chickory coffee.
Go to Jean Lafitte's blacksmith shop on lower bourbon St. (go away from Canal St. past all the big clubs like Pat O'Briens and Cat's Meow, and past the gay dance clubs. ) Speaking of Pat O'Brien's, lots of people go there.
Also, be sure to get to Preservation hall Jazz club before you start drinking one night to enjoy old style New Orleans Jazz. Probably the only place where you'll hear authentic old style.
If you still have time and money, you can check out the bywater. Look at going to Vaughn's. Take a cab, and take a cab back to the quarter / CBD. I would walk or bike from the quarter, but you don't know the area and it is not always safe. The bywater is very funky. May or may not be your thing.
St. Roch Market is a great food stall and has excellent happy hour cocktails.
If you want good beer, go to DBA on Frenchmen Street.
There's actual Absinthe at the place on Pirate's Alley between the cabildo and the cathedral off of Jackson Sq.
For a good breakfast, go to Cake Cafe in the Marigny (past elysian fields from canal). I think it is on Decatur, but I don't recall. There's also Elizabeth's out in the bywater.
Tipitina's is a famous music club.
There are a few things in there that didn't involve eating or drinking to do during the day. Some people still want to go on the disaster tourist stuff. If you want to rent bikes, go to my friend Bicycle Michael's on Frenchmen St. Tell him you know me. He'll probably say something like, a lot of people know Chris. ;-)
Tune in to WTUL, 91.5FM. It is Tulane's college radio station. They've got club and event listings hourly, and you can win tickets pretty easily from them for shows. Also check out WWOZ, 90.7 IIRC. They are NOLA cultural station. They also have club and event listings.