New Orleans is one of the most fun cities in the U.S. You are almost guaranteed to have a great time, but with that in mind, I have a few suggestions...
Take advantage of the arts scene. The arts and architecture are my favorite parts of the city. There are tons of free galleries downtown, so for an awesome afternoon, go gallery hopping on Royal Street. On top of that, look up and pay attention to the architecture. Spanish architecture has a profound influence on the character of the French Quarter.
Regarding Alcohol...It is legal to carry open containers of alcohol around New Orleans (though technically they're supposed to be in plastic/styrofoam cups). That being said, public intoxication is still a thing and a lot of people get arrested for it during tourist season. Drinks in the French Quarter cost around $8 - $10 each. If you want to just walk around observing the neighborhood while enjoying the open-container rule, I suggest buying alcohol from convenience stores. Unlike most places, convenience stores in New Orleans sell liquor in all sizes.
Do not drive anywhere. If you drive to get to New Orleans, park your car somewhere on the street and leave it there the entire time. It's a pretty walkable city, I recommend bringing or renting bikes. I used A Musing Bikes and had a great experience. You can easily see and traverse the city on bikes, and you get to pay more attention to the beautiful architecture. The public transportation is also very easy to use. Right when you step into a bus or trolley, you can buy a $3 pass that's good for unlimited rides for 24 hours.
Hostels are your friend. There are tons of hostels in New Orleans, and since it's a touristy city in the Spring and Summer, hotels are way too expensive (does anybody stay in hotels anymore?).
Leave the French Quarter. My favorite experience in New Orleans was riding my bike randomly around the Garden District, where beautiful houses and old cemeteries are covered by huge overhanging trees. The Warehouse District has wonderful, quiet restaurants and galleries.
Do not pay covers to get into clubs. Clubs downtown are pretty much empty 2nd story rooms with loud music and a bar, so if you have to pay a cover, I promise you can find the exact same experience for free within 100 feet.
Do not go to Bourbon Street every night. Yes, it's fun to see the mayhem there, but it will get old. Check out dance venues elsewhere in the city for places with more character, or go on a night tour to complement the city's spooky character.