For my FI's 21st his parents took us to the quarter for the night (we live in the 'burbs of NOLA so it was easier to get a hotel room) and we stayed at a place called the French Market Inn. It was right across from the Jax Brewery - really convenient.
If you're looking to come down for NYE, book soon. With the Sugar Bowl, things book up hella fast.
Places to eat:
Camellia Grill - it's on Carrolton, on the streetcar line. Best damn greasy spoon ever.
Maspero's - in the french quarter, across from Jax Brewery. Amazing baked ham and cheese sandwiches and $2 strawberry daquiris. Beware - they only take cash.
Cafe du Monde - Cafe au lait and beignets. Yum!
If you're looking to spend more serious cash, try Commander's Palace, Brennans or Galatoire's.
When you're in the FQ, go to Pat O's piano bar. Have a hurricane. Remember that there's a $2 deposit for the glass - if you don't return your glass at the end of the evening, you won't get your deposit back and you keep the glass. After FI's birthday party, we ended up with 6 of them.
Don't ride the city busses. Just don't. The streetcars (its not a trolley, dammit, its a streetcar!!!) are pretty safe and convenient, but don't run all the way into the quarter.
Take a ride through the Garden District in the daytime. It's gorgeous. Note the big church on Loyola's campus, right on St. Charles - its where I'm getting married!
Have fun. NOLA is a great city.