Some things that you should try to do while visiting New Orleans, especially if you want a more authentic visit outside of the French Quarter:

The National World War II Museum (formally the D-Day Museum) is a must-see, they are located in the CBD or Warehouse District area on Magazine Street.  While in this area, also visit the Ogden Museum of Southern Art and the Confederate (Civil War) Museum right next door.

Walk along the next side street up and down Julia Street and visit the local (not tourist trap) art galleries.

Then head up Magazine to the Garden District and Uptown New Orleans for the most unique and fantastic shopping, lots of small quirkey shops for all budgets mixed in with great little eateries, more art galleries, antique shops etc.

Move up a few streets to Prytania to see some nice old Victorian and Georgian homes and then of course, up further to St. Charles Avenue to the grand mansions and estates of the Garden District.

Food places to check out:

Martinique Bistro in the 5900 block of Magazine Street, reservations needed, ask for a table outdoors.

For a quick but good quality and reasonably priced meal, go to ZEA Restaurant on St. Charles Avenue.

For cheap and good eats - go to Juan's Flying Burrito on Magazine (notice the trend here, MAGAZINE Street!)  Pizza?  Reginellis guessed it, Magazine Street.

Want great Mojitos?  Bridge Lounge near the Crescent City Connection Bridge.  Good BBQ? BBQ Mama's next door to Igor's (great local bar hangout) on St. Charles Avenue.  

Lingerie, go to Basics Underneath on Magazine or Trashy Diva on Magazine.

Looking for a GREAT place to stay?  Try this:  Creole Gardens Bed and Breakfast, they are the greatest bunch of people and the place is very cool.  Not right in the middle of stuff but close enough to get to everything in 5 minutes.  If they can't help you out, try The Garden District Bed and Breakfast right down the street. You can find reviews about both places here on Trip Advisor.

If you stay at Creole Gardens or Garden District B and B and want to go to the French Quarter, as a local, take a CAB!!! It is the best way to do it because you won't have to worry about parking costs, finding a place to park, getting your car broken into AND if you are planning to take in some nightlife, you don't have the worry of drinking and driving.  United Cab is quick and reasonably priced.

Tattoo while in town?  Call Walt Clark with NOLA TATTOO in Uptown near the Riverbend.

Want a massage while in town?  Go to EarthSavers Spa OR BellaDonna - both on Magazine.

Want to bring back something unique for your friends?  Don't buy tourist junk in the Quarter, go to Blue Frog Chocolates Uptown on Magazine for home made chocolates in New Orleans shapes, they also have other local treats and cute gift items.