For the French Quarter I HIGLY recommend the Hotel Monteleone. Very nice place a block from Bourbon St.
I'll get some good recommendations for the Garden District. I like the Hampton right next to Audobon Park as it's a beautiful area right near Tulane, Loyola, the Zoo, all of Audubon Park and just some great restaurants and really the Uptown New Orleans experience close by, but it's a bit far from the quarter and the stadium. I'll ask my wife what her thoughts are. Are you and the wife more into nightlife, all about the history of the city, wanting more of the food and culture type experience, etc?