For a crash course in Final Fantasy, find the GBA Dawn of Souls.
Play it. 1 because that's just mandatory, and the GBA version is the most... modern-gamer friendly. Then play 2 to get a taste of how all the proceeding games will be set up.
I suggest you proceed with nine because it is a good representation of the modern final fantasies, then choose something between three and six. If those stole your heart, play either FF4 or FF6, depending on wether you had played the other or not. Only then is it safe to tread into the, "Modern" final fantasies, but I suggest playing more of the pre-seven games first. You must choose very carefully with modern games, because whichever you play next will determine your taste in the series. FF8 is a more serious game, featuring a pretty believable world with rediculous plot devices, like flying school battles.
Final Fantasy seven and ten appeal more to an anime fan; if you like shows like Fruits basket, go for 10, if you like things like Naruto or Bleach, go for seven.
If you chose eight, proceed to play seven then ten.
If you chose to play seven or ten, play the other, then play eight. If you liked seven and ten, you will likely dislike eight, so don't make a big huff about it.
FF12 is a good all-around game, and is a good pallete cleanser after the strong tastes of its name-only predecessors.