I grew up in NOLA and have been to 30+ Mardi Gras. IMO, you need to go to one night parade in downtown. That is the true experience.
Catch Endemyion or Bacchus on Poydras street. Safety is not a huge concern if you don't go on the dark back streets. Stay on the main drag and in the crowds. Don't go bar hopping after the parade, but do walk the French Quarter/Jackson Square and even Bourbon, during the day. Get a moufelatta at Central Grocery, a beignet at Cafe' Du Monde, and watch the street performers all around Jackson Square. Walk through the French Market too.
One parade is enough to say you've done it! However, I would spend Mardi Gras Day at the truck parades that follow Veterans Blvd. in Metairie. That's where locals bring their families. The parades run from just about sun rise to sun set. Bring your chairs and food for the day.
With those two experiences, you'll have a good snapshot.