Wow.
Let me begin by confessing my jealousy of those fortunate enough to be in the Old World, not just in France but all over. The world of incredible cheeses, which must be sought out and pursued like a choice second-hand-store find here in the USA fills the streets and markets there. It is as much a part of the atmosphere as the wide boulevards of Bucharest, the yellow-hued walls of the old quarter in Palma de Mallorca, or the gardens of Berlin. You breath it in until it becomes another necessity of life, a staple without which your diet would crumble.
To make a list of cheese to try when in France is a tall order, and I am certain that anyone who has lived in France will have tried many more French cheeses than I ever have, see jealousy. To even presume to know the world within a world within a world that is French Cheese fully enough to give a comprehensive primer is beyond even my snobbery, but here are the cheeses I would seek out with haste and anticipation upon arriving in France:
Goat Cheese
French cheese would not be the same without chevre, and there are a million different brands, shapes, sizes, and flavors to choose from. I would say start with the plain, look for anything au lait cru, and then branch off into that which is covered in ash, or paprika, or juniper berries, herbs de provence, etc etc.
Definitely find a Crottin de Chavignol, for while it is a little more potent it is also super delicious.
Brie
Oh dear, the Brie. Brie de Meaux and Brie de Melun are pretty safe bets for life-altering, world changing cheeses, but if you're in Paris you will have no shortage of Bries vying for your Euros no matter where you go. My advice? Wait for the the spring and summer (these are young cheeses, buy one in January and it's from at the latest December milk... which means December food for the cows.... which means hay. You want the real pasture and twigs and mushrooms and such flavors of warmer weather) months and ask the stout yeoman for three cuts of brie at three different ages. That plus a bague...
Sorry, daydreaming again.
Also, Reblochon is a game-changer. Try some.
Sheep Cheese
Again, there are so many options, and mostly all are great. See if you can find Causses d'Argental or anything from the Pyrenees Mountains, anything from Jura, anything Basque. It won't disappoint.
Blue
You can find a lot of the great French Blue cheeses here in the States, but if possible look for some stranger sounding Blues, or anything that is the "reserve" or "superior" etc variety, as that will likely not be available at home. I found a Roquefort Superior while I was in Germany... preeeeeeeetty much fell in love with Roquefort again.
Swiss
Beaufort and Comté have my heart when it comes to French made high Alpine cheeses, and they're also very approachable for the less adventurous among us. Sometimes when I say approachable I mean "lame and boring" or "for the unenlightened", this time I mean "mind-blowingly delicious but not stinky".
Stinky
These, unlike some of the cheeses I have listed above, you cannot so easily get at home, so when you're feeling bold search them out: real Munster, real Époisses, real Livarot.
Overall you should look for that which you cannot find in the US, assuming you, dear reader, are from the US. That will usually mean looking for young (on the scale of weeks or up to two months) and unpasteurized cheeses, or anything that just looks completely foreign. If you see something that you've never seen before, ask for a taste of it. Any self-respecting stout yeoman will be happy to provide a sample. If you like it, buy it. Put a ring on it. Pick up some 4 Euro wine and a fresh baguette, and have a ball. Like I said, this list is a little hurried and far from complete, but the good news is that the best education comes from trying new cheeses first hand.