I have been to Mardi Gras many times. My best advice is to stay away from Bourbon Street for anything more than one obligatory stroll just to see what its like. And ... ah, you're gay ... then you will want to see Bourbon St. between St. Ann and Esplanade, not between St. Ann and Canal.
The parades are awesome. Be sure to catch at least some of them. I don't know how long you will be there. By the Thursday before Mardi Gras (tonight), parades will be rolling pretty much non-stop. Endymion, on Saturday night, is the biggest (the most throws and floats) and is pretty family-friendly. (Actually, the only parade that is not family friendly is Krewe de Vieux, and it rolls through the Quarter several weeks before Mardi Gras.) Endymion rolls through Mid-City. Most of the other parades are on St. Charles and Canal. Bacchus -- another big parade -- is Sunday night. Proteus is an old-line parade on Monday night. They still use the old wagons from the 19th century, and so the floats are smaller and they don't throw as much (the wagon can't support the weight of the throws). I find Proteus the most beautiful -- the floats just shimmer as they go down St. Charles. Proteus is followed immediately by Orpheus, another big parade. Zulu, a predominately African American krewe rolls early on Mardi Gras morning, followed immediately by Rex, which is the most most prestigious/formal of the parades. Zulu is a riot. It's completely politically incorrect to an outsider (you'll see), but I love it. Rex is probably worth seeing once -- it's pretty. The issue with parades on Mardi Gras day is that there is so much other stuff going on that the parades may not be the most interesting options ...
I usually head to the Backstreet Cultural Museum in the Treme early on Mardi Gras morning. I grab some red beans and rice for breakfast there and hang out with skeleton gangs and baby dolls. If you're lucky, you'll see some Mardi Gras Indians, too. (Mardi Gras Indians are absolutely amazing!) Late morning I head toward Frenchman St. and the Marigny. You will usually see the best costumes around the R-Bar -- at Royal and ??? (I don't remember the crossing street) in the Marigny.
Oh, and be sure to come up with some kind of a costume for Mardi Gras day. You will fit in much better with a costume.
And I almost forgot ... Zulu sponsors a festival at Woldenberg Park, on the bank of the Mississippi on Lundi Gras (Monday afternoon). It's a nice place to hang out, and they usually have some top notch musicians. And it's free.
You should be prepared for any kind of weather. I have been there when it's 75 degrees and sunny, and I have seen it 35 degrees and pouring down rain (which made for a very miserable Mardi Gras).
One more thing --- the local news stations have free apps with parade maps. Grab one. The parades don't always run on time, and the apps will have up to the minute info.
Museums will be open through Sunday. The Louisiana State Museum might be open on Monday -- I'm not sure. All museums --- and many (but not all) restaurants --- will be closed on Mardi Gras day.
I'm jealous. Have a great time! I usually go every year, but I am headed to Egypt in a few weeks. I couldn't do Mardi Gras and Egypt back to back.