Man... This might get long :P

Parades:
[parade site]
http://www.mardigrasneworleans.com/schedule.html
Click on any parade and there's a link to an interactive map. Check there to see where all they're going and find somewhere you think you can get to. Good luck driving anywhere near the parade route, by the way. The entire NOPD is on call for every parade.

Thursday:
If you make it into town for 615 or shortly after, Muses starts 6:15. The parade is all women and is usually pretty entertaining.

Friday:
I think Krewe D'etat is the big one for Friday. They've been voted best parade a couple times, although it's not as big as the super krewes. They start at 6, and if you play it right you can end up pretty close to the French Quarter at the end of the parade and just segway to Bourbon if you so wish.

Saturday:
Endymion is one of the three 'super krewes', so I would go to if I were you.

Sunday:
Bacchus is another super krewe, over a thousand riders + celebrities and what-not. Not to be missed :P

Should try to find a way to stay for Monday and Tuesday... The third super krewe, Orpheus, is on monday, and then Zulu and Rex are some of my favorite parades because they usually have cool throws (Zulu has coconuts! And Rex usually has a pretty cool bead every year) Then there are two more parades that follow Rex.

General parade advice:
mrtr covered a lot of it...
Be reasonable afraid of anything that touches the ground. You'll be here for the earlier parades, so it won't be too bad, but the ground is still... Eugh... Their idea of cleaning the streets is a street sweeper and some water.
I dunno how you feel about temperature, but it'll probably be in the 60s for most of the parades, but it tends to cool down pretty quick once the sun sets, but I think the lows for this weekend are like mid to upper forties... hardly cold, especially for a yank ;)
Don't wear anything to a parade that you would be upset if it got ruined... or didn't make it home.
Carry as little in your pockets as you can get away with. ID+cash is plenty. Parades are a hot spot for pickpockets because you're getting hit by people constantly and won't be as likely to feel them as they nip all your stuff.


Foodstuffs:
Cafe du Monde is a good place to go... They sell coffee with chicory which is pretty hard to find in most places. I don't personally like it, but I know people that live on the stuff. Cigar Factory is down the street from them, although I've yet to find another shop in the French Quarter worth anything. Most of the "cigar shops" you'll find on the internet are more like head shops. Oh well.

I'm a bit hesitant to recommend anywhere for food during carnival season just for the fact that it's so hard to get in anywhere. Eat whatever you can get your hands on, really. Try etouffee while you're here if you get the chance, preferably with crawfish. If you go anywhere to eat, ask what the house specialty is of the restaurant and order that unless you're allergic to it (yes, even if it involves squirrel or snake... Or neutria (gigantic rats) :P) But it won't, most likely. Just tons of seafood.

If you've liked to learn how to work hard for a small amount of food while crying from the spiciness (and aforementioned spices working their way into small cuts on your fingers) order boiled crawfish somewhere. 3:)

General advice:
Cops are going to have more stuff to deal with than you, most likely. Don't do anything too stupid and it won't be too bad. The number of cops is supposed to serve a deterrent more than anything else.
If you have to carry your wallet with you, wrap rubber band around it (Or if you have a junk wallet, put velcro on your wallet and velcro in your pocket. It's pretty hard to miss someone ripping the velcro out of your pocket.) Keep it in your front pocket, either way.

If anything else pops into this random head of mine I'll post it...