Restaurant Iris (NOLA native), Anthony and Michael's Italian Kitchen, and Hog and Hominy
Three awesome places. Andrew/Michael chefs were just named best new chefs by Food and Wine magazine. A/M is fine dining. Their more casual joint (which is incredible) is Hog & Hominy - it was just name one of the 12 most outstanding restaurants of 2013 by GQ magazine. These boys are on a roll.
As far as BBQ goes - it's already been said in here, but BBQ Shop and Central BBQ are the two go to spots these days.