Answering your requests more specifically. My work location and apartment were in Metairie and Lakeview suburbs, but most of the places I list below are near the French Quarter or the area called Uptown (which you should not miss).
Latest and Greatest?(see list above)
Mondo: this is a Susan Spicer place in Lakeview, which I always enjoyed. Closer to where you will likely be staying is her Bayona, which gets raves.
SoBou: that's short for "south of Bourbon", see Brennan. Meh, not my fave
G. W. Fins: upscale seafood, we liked it
Coquette: Uptown area, most excellent brunch, must return for dinner
Classics?Mr. B's: a Brennan place, excellent, just across street from the Monteleone – see below
Acme Oyster Bar: there are at least two of them
Felix’s Oyster Bar
Court of Two Sisters: brunch (natch)
Central Grocery: for a muffaletta (natch)
Galatoire's: dress code requires a jacket, they’ll provide one if you don’t have. Despite the required jackets, it's not stuffy or formal.
Drago's: seafood, I always went to the one in Metairie, but there is one in the Hilton on Poydras, must try the chargrilled oysters
There are many others (Antoines, Commanders Palace), but these are the ones I visited.
Drinks?Tableau (see Brennan): This is a restaurant but they have a nice bar – lots of marble, very good
Sazerac Bar at the Roosevelt Hotel: classic cocktails, beautiful hotel
Carousel Bar at the Monteleone Hotel: classic II, don’t count on getting a seat at the carousel bar, the seated patrons rotate around the bartenders.
Razoos: if you want a little more rowdy, it was 4-for-1 beers when we went a few years ago, more recently with the focus on moderation, it was cut back to 3-for-1
Attractions?La Boulangerie: Uptown. A very authentic French Bakery, really great
WWII Museum: very well done and huge
Mardi Gras World: a museum and working facility, sounds kind of hokey, I guess, but I found it fascinating. It’s the place where they craft many of the very intricate parade floats
Landry's: this is a restaurant on Lake Pontchartrain, which will be a 20 minute or so drive from the French Quarter, the attraction is the setting, the water and the sunsets. Food (seafood) is OK
Saenger Theatre: beautiful restored theater (think amped-up Palace). See what's playing during your stay.
http://www.saengernola.com/mobile. The Sazerac Bar at the Roosevelt is within walking distance.