We went to NO last summer without the kids, but we'd love to go back and bring them with us. We went last June, and it was HOT. I grew up in Houston, I've spent plenty of time in Florida, but there was a heat wave going on at the time and the heat index was around 110 the entire time we were there. BE PREPARED TO SWEAT!!! Drink tons of water!
You said you have teenagers. Do y'all like jazz music, or at least live music? To us, that's a must-do. Preservation Hall is where Harry Connick Jr. got his start in the music business. They don't serve food or drinks (though you can bring in your own bottles of water), and we saw very young children there. AMAZING music!

But....NO AIR CONDITIONING.
I know we also saw kids at Palm Court Cafe. It's a restaurant that also has live jazz music.
The ghost tours/Haunted History tours are fun, and teens would probably really enjoy those. We went on two different tours by Bloody Mary, and those were fun.
Definitely go to Cafe Du Monde and get beignets! You can either wait and sit in the cafe (good for people watching), or go to a walk-up window and buy your food to go. We always got ours to go, then sat in the park across the street to people-watch. Be sure to shake up the beignets in the bag so they get good and coated with powdered sugar! Yum!
Central Grocery (half a block from Cafe Du Monde and across the street) is the birthplace of the muffaleta. I can't stand olives, and I STILL loved this sandwich. Good food and cheap.
If you can afford it, try to splurge on one meal there. Go for lunch if you need to cut costs further. Whatever you do, there's no excuse for eating bad food in NO!
A word about safety...
We are friends with two couples who grew up in NO, and all of them said just to be very careful at night. They said the Quarter is safe enough, but as soon as you get to the edge -- take a taxi back to your hotel, even if it's only a couple of blocks away. Now, we were staying just outside the Quarter, so we weren't driving around. We saw big parking lots on the south edge of the quarter, near the Aquarium, and those were probably safe enough to walk to until...I dunno, maybe 11:30 or midnight? The more respectable-seeming places that focused on live music seemed to shut down around 11, so that's when we decided to leave each night. Also, our friends told us that after 11pm or so, Bourbon St. was the safest way to walk to the edge of the quarter.
I know a lot of people won't take kids on Bourbon street -- for good reason. However, with it being NO, you never know when you might see something...inappropriate.

I'm sure I'm in the minority, but I'd take my kids down Bourbon St. at night *if* it's the safest way to get back to the hotel. I'd just tell my kids that they'd likely see all manner of inappropriate and/or stupid behaviour, that even adults can behave worse than kids sometimes, and they were to stay with us, eyes forward, and just KEEP MOVING. Hmmm, it could be a teachable moment even: "See kids? Getting drunk makes you look really stupid! Why would anyone ever want to do that???"
Good luck and have a great time!