I would second Commander's Palace, but suggest brunch over dinner. While you are up that way, walk around the Garden District and enjoy some of the cool houses. There are also some very cool shops and restaurants on Magazine St (on the border of Garden District and Irish Channel). I'm a big fan of the Bulldog which is a great beer bar with an awesome outdoor patio.
Frenchmen St is a good spot for a lot of music options in one place, but can still be a little touristy. For more of a local scene, I really like the Maple Leaf (on Oak St) for live music. As a bonus, just next door is:
http://jacques-imos.com/ which has amazing food. Do dinner there, then music at the Leaf.
The Cafe du Monde in the quarter can be crowded and annoying. If you just want to sample the beignets, they have an outpost in the River Walk mall. Same food and you can sit out on the balcony and watch the river.
I'd suggest walking around the French Quarter during the day. There are a lot of cool blocks that are not on Bourbon St (e.g. between Bourbon and the French Market). Grab a beer or a cocktail and wander around looking at cool buildings and window shop at antique shops. I'd second Lafitte's (it's a dive bar, but the history of it is cool); it's far enough down Bourbon that you won't get hung up in the "show your t*ts" crowd.
In terms of dining in the Quarter, the two iconic places I like are Arnaud's and Brennan's (former for dinner, latter for brunch). Both are pricey though.
I also like Acme for oysters, although if the line is too long there, there is a place right across the street is a good substitute (Felix's).
Have a great trip! It's one of my favorite US cities.
