New Orleans has some great bars and music venues such as Tipitina's, Mermaid Lounge, Maple Leaf and the Rock and Bowl to name a few.
Eat at some nice restaurants. New Orleans has some world-class places that feature the distinctive creole cuisine from the area. Try the gumbo, the jambalaya and other local favorites.
Have a beignet at Cafe du Monde (it's touristy but is still the best place to catch a beignet).
Go to Harrah's casino...it's very nice, not crowded and has everthing you can find in Vegas except a sports book.
Go for a run along the Mississippi River.
Quick Tips:
Get out of the French Quarter as much as possible. It is fun for a couple of days to get a feel for the place, but New Orleans has more interesting things to see and do after you've walked around a little bit. If you do stay in the French Quarter, the side streets (i.e. Royal St.) are much better than Bourbon St. (unless of course you enjoy hanging out with a bunch of drunk, loud people...I do sometimes).
Avoid the buffets in the French Quarter. They are all terrible for the most part.
Best Way To Get Around:
Don't worry about getting a car. If you stay in the French Quarter, you will do a lot of walking. You can use the streetcars to see sites like the Garden District. And tour companies will drive you to the sites outside of the city. Your tour will cost a little more, but much less than if you had a car for the week (and that doesn't even take into account the hassle and expense of parking). The St. Charles Street streetcar is a great way to see the Garden District (try it during the day and at night).