Definately go to the French Quarter during the day. It's beautiful. There is an open market area. Eat at Cafe Du Monde. They have the best coffee and benets. Though the line to get a table is long, the wait is worth it. Bourbon Street is also in this area, but there is more to the French Quarter than just a party street.
The city park is also a lovely area that shows part of the New Orleans soul. They have a great golf course, nice running areas, and just beautiful sights. Try the Garden District. They have glorious homes. Astounding gardens, and you will be in love at first sight. If you like art, museums or trendy restaurants, try and visit the Warehouse/Arts District. If you adore shopping, try Magazine Street. It's a shopper's dream.
Even though you are missing Mardi Gras season, you can still enjoy some of the heart of a fun place. Thanks for asking a great question about my hometown :)