We did the Honey Island Swamp tour. It was fine, but I think the aim was to see 'gators in their natural habitat. We didn't see any (wrong time of year possibly) but the boat driver brought along a baby one..a bit forced perhaps. But it is a great
way to get a look at the swamplands, and to get a glimpse of how people live in that area. We enjoyed it, would recommend it for that. You may even see a few 'gators when you're there.
Definitely stop in at the Cabildo (museum) on Jackson Sq. It's part of the Louisiana Museum. They always have some great exhibits, and if you're a history buff, the permanent collections are really worth a look. You could easily spend at least half a day there. Louisiana State Museum
The best way to see the FQ is to simply stroll around, there is so much history in the buildings that are utilized for everyday purposes. If you find yourself in an old bar or store, have a chat to the staff or regulars about its' history. New Orleanians are very friendly, and happy to chat!
There are also many walking tours available; I'm not a walking tour type so cant tell you which ones are worthwhile. I'd recommend a carriage ride when you first arrive..yes, it's very touristy but you get your bearings. You also get information, but be aware the driver may (or may not) embellish the facts somewhat. This possibly also applies to the walking tour guides.
I think you should take a walk down Bourbon St, but the only real venue of possible interest there is Fritzls Jazz Club. It's a cosy little place, groaning with musical history, and a good spot to hear traditional jazz. If it's your cup of tea, Preservation Hall is another jazz spot with a lot of history.
New Orleans is a wonderful biking city. Hire a bike and have a look around the Garden District, or bike around City Park. Just be aware the roads, while nice and flat, are a little...under-maintained.
If the heat is a bit off-putting for biking, it's pretty cheap to hire scooters as well. We rode around the Ponchatrain a little bit, but there's a lot of roadworks (!) around the foreshore, so it's hard to see a whole lot. That may have changed though.
As with any town, have your wits about you and stay alert, but don't be paranoid. Don't walk alone too much outside heavy foot-traffic areas late at night. Cabs are plentiful at that time, so don't feel you can't go out at night. The city is especially magical after the sun goes down.
Have a great time in my favorite town!
EDIT: Cemeteries! How could I forget them?? You HAVE to see at least one during your stay. There are tours, or hire that scooter and explore at your leisure.