You need to go to one of the PILE (Poste d´Information de la Légion Ètrangere) on the french territory. You will pass a first medical visit there, a very short one. You will stay there for a few days, sign your first contract, then you will be taken to the 1er RE (Regiment Etranger), for the medical, psychotechnical and physical tests. On that time, you will be a EV (engagé volontaire) also called "bleu" (blue) because as long as you are not aproved, you will have to wear a blue shirt. The psychotests are at a computer, in every language you need. The physical tests are the "beep test" (never did it, it is new), to climb a 6 yards rope twice, without using your legs, and some pull and push ups (oh my god, you will make a lot of them in the next months), The med tests are pretty serious too.
If you pass all tests, you will be taken to Castelnaudary, to the 4eme regiment etranger, for your basic training (16 weeks), There, you will learn how to be a legionnaire, learn french basics, and after a month, you will have a march of around 80 miles with around 30 Kilograms of Paquetage, your FAMAS (french assault rifle) and if you do it in 2 days, you will be allowed to be part of the french foreign legion. You will be allowed to wear your képi blanc for the first time, and say your codex in front of your commander.
My advice: stay at home, my little american friend. I saw a few former marines give up and run away. We are not talking about the air force here. There are a few things you can´t be prepared for.