I live in NOLA, so I'm happy to answer your questions. The Hampton on St. Charles is just fine. Nothing fancy, but I've had friends stay there and say it's clean and comfortable. That area is just down the street from the big, fancy Garden District mansions, so you'll definitely want to see those. You can either walk around on your own or take a tour. I recommend Haunted History, they are in my opinion the best-researched tours in the city. That would be their "Homes of the Rich and Famous" tour.
https://hauntedhistorytours.com/our-tours/garden-district-tour-new-orleans/ If you have any interested in the ghost/vampire/voodoo history of the city, take a look at their full tour list--ALL of them are worth the time and money, so just take your pick.
You'll also want to visit Lafayette Cemetery, which is just a few easy walking blocks from your hotel. It's currently closed for repairs, but will likely be open by your trip. If not, you can still see quite a bit of it through the gates. We're known for our elaborate above ground cemeteries (Cities of the Dead), and that's one of the most stunning.
You may also want to dine at Commander's Palace, which is across the street from the cemetery. It's been a landmark since 1893 and has launched the careers of innumerable chefs, including Emeril Lagasse and Paul Prudhomme.
You'll be on the streetcar line, so DEFINITELY make some time to take the streetcar to the French Quarter. It'll end on Canal Street, and then you just have to cross that street on foot. Bourbon Street is one of those must-see tourist places, but you won't want to spend much time there. Gawk at everything, take a few pictures, and then get off it again. You could easily spend your 6 days just strolling through the French Quarter, so leave plenty of time to explore. If you're into high-end antiques, you'll want to walk down Royal Street. If you're into vintage stuff that hasn't been all polished and shined up, head to Decatur Street around the intersection of Barracks. If you want to visit voodoo row, head down Dumaine Street to the voodoo temple, museum, and shops. Be sure to visit the French Market, which still hosts a produce market on the weekends along with a huge flea market that's open every day.
Also make time to visit Jackson Square, which is the literal and figurative heart of the French Quarter. You'll find visual artists and mimes during the day, and tarot readers at night. There are also three museums along the Square, as well as the St. Louis Cathedral, which is the oldest still operating Catholic cathedral in the US. Be sure to get beignets (French donuts) and cafe au lait (coffee with chicory and milk) at Cafe du Monde across the street.
Otherwise, it just depends on what you're into. We have tons of museums and tourist attractions of every description, incredible food and drinks on every corner, live music in every bar and on the street....
As for transportation from the airport, I'd take Uber/Lyft. It's quick, easy and reliable, and not very expensive. Other than the streetcar, you can get around on foot the rest of the time. The city is very walkable. Use street smarts, stay close to other people, carry cash but don't flash it around, and you should be fine. Oh, and if someone bets they can tell you where you got your shoes, tell them "on my feet right here in New Orleans" and keep walking. It's a common scam.
Let me know if you have any other questions!