Are there any other districts worth checking out besides the French Quarter?

If you're staying in NOLA for a week, definitely don't spend it all The French Quarter; venture out!  Though the French Quarter is very stereotypically New Orleans, it's also very small (not to mention very touristy) and you can very likely see everything there in a day.  You can walk through the streets and look at the historical buildings, sample some restaurants, do some souvenir shopping, walk along Decatur, check out the famous landmarks, etc, in the day time.  Then come night time, you can head to Bourbon Street and experience some of the nightlife there.

Over the next few days, here are some other areas that I suggest checking out:

• Frenchmen Street -- this is a must-do for a first time visit to NOLA!  This is THE place to listen to jazz.  Hit it up on any night of the week for some live jazz at a number of venues.
• Magazine Street -- Long street that spans a couple neighborhoods.  Tons of great restaurants, bars and various types of independent shops.  Great for antiquing and finding unique gifts!
• Lower Garden District -- Lots of beautiful houses to look at.  Magazine Street runs through it, so that's a good one to walk through.
• Central Business District -- It might seem a little dead at night, but it's actually home to a lot of must-go-to restaurants in the city, like Cochon and Restaurant August.
• City Park -- One of the most popular parks in the United States.  It's beautiful and it's huge, so there's a lot to explore.  There's an amusement park for children inside of it, plus a Café du Monde-esque cafe called Morning Call.

Debbie is absolutely right. It would be well worth your time to spend the day taking the streetcar up St Charles Ave to see the Garden District and Uptown. There are beautiful antebellum homes, two university campuses, Audubon Park, and plenty of watering holes (ie. Avenue Pub - best beer selection in the city). A bus ride up Magazine Street to the same areas is another option with plenty of restaurants, bars, and shops.

Outside of Bourbon Street in the FQ, here are a few cool local/less touristy hangouts: Molly's at the Market (1107 Decatur), Erin Rose(811 Conti), and the Chart Room (300 Chartres). Frenchmen Street should be your go-to night out spot with numerous music venues to choose from.

I'd be happy to give more specific recommendations based on a visitor's taste. Just ask!

Marigny/Bywater= Frenchmen Street. Walking distance to the quarter. best live music in the US hands down

I would take a streetcar from Canal St. (borders FQ) to the Garden District. Get off near the Washington stop. Walk through the neighborhood to admire the mansions on your way to Lafayette Cemetery No. 1. Get a guided tour with Lafayette Cemetery. This is a smaller, more unkempt (spooky) cemetary than the more famous St. Louis Cemetery No. 1. After that, you can have lunch at Commander's Palace where they may have 25 cent martinis at lunch! Or browse the cute little Garden District Book Shop. 

It's a small little excursion, but gives a great feeling totally different than the French Quarter. Enjoy!