Each main game is standalone - there's no overlap. Pick up one of those (excepting XI or XIV, which are MMRPGs) and give that a shot. The first six are more considered the oldschool ones, focused a little more on combat and gameplay mechanics. From 7 onwards, they became more....interactive novels, focused on the story. X and XII are both "love 'em or hate 'em" situations, with the story in XII being a tad more hidden, but the production (voice acting and dialogue) being superior. The original Playstation games' graphics are a tad dated at this point, where 7's generally considered the best (aside from graphics), VIII is liked by people who were new to the series with 7, and 9 appeals more to people who enjoyed the older games.

My personal recommendation would be, if you have a DS which supports Gameboy Advance carts/GBA, pick up FF IV, or VI for the GBA. It'll give you a little more idea what the series is like. After that, just jump into whichever one you feel like doing. If you like the oldschool grindfests, look into FFI+II for the GBA/FFIII for the DS. If you like the story portion, look into the Playstation games.

As for the spinoffs, most of them switch genres. Avoid the original Gameboy Games/Mystic Quest - there's really no point. If you're a fan of tactial RPGs, give the Tactics games a shot. If you play and really enjoy VII, IV, X, or XII, their direct sequels would probably interest you. Ignore Dissidia unless you've played and enjoyed a lot of the games (it's a fighting game featuring one hero/one enemy from each of the main series through XI). Don't bother with any of the mobile phone games. Crystal Chronicles games (excepting the WiiWare ones and the upcoming Crystal Bearers) are multiplayer games whose fun depends on the number of people playing - if you can't get 3 or 4 playing, don't bother.