Great museum! Especially good for a bad-weather day since it is all indoors. If you want to see everything plan on spending the entire day there!
Fun place to wander. If you are there at night stay on the main roads where everyone else is. Don't go off on deserted side streets. Otherwise it's very safe.
Great beignets! Get an order of beignets (3 per order) and a hot cafe au lait. It's self-seating so when you see someone getting up snag a table! Cash only. Don't wear a dark shirt and don't inhale - that powdered sugar will be all over you! :)
2 mile paved loop in the park is a great location for a run! Or run up/down St Charles in the street car tracks (run towards the street car so you can see them coming).
Bike rentals including Brompton folding bikes!
Try Viator.com - I haven't done any of these specific tours but have used them in other cities. Some good ideas include a French Quarter walking tour, Garden District walking tour, food walking tour, cocktail walking tour.
There are several available. Most are a little ways away so plan at least half a day if not a full day for this. A couple of popular ones include Oak Alley Plantation, Whitney Plantation, Nottoway Plantation.
Multiple tour operators available - I don't know enough about them to recommend any specific operator.
Take a Street Car ride up/down St. Charles Avenue and check out the beautiful homes the line this grand avenue. You need exact change but it's only $1.25. I think there is also an RTA app but I haven't used it.