Settle down, this might be lengthy.
I've been a Final Fantasy fan now for more years than I conveniently count. The series has its high and low points, and your own individual tastes will count for a lot with regards to which games you particularly enjoy.
Personally, my favourites are VII and VI. I've poured hours, days, weeks...
years into these. I still play them today, and I'm reasonably certain that I'll still be playing them another 10 years down the line. Characters, story, music, villains - I'm just absorbed by these worlds. The combat mechanics are quick to pick up and enjoyable to tinker with. Though as a fair warning, you'll be able to go make and eat dinner by the time Knights of the Round has finished a quadra magic summon in VII. Maybe I'm one of those rabid fangirls, I don't know, but this game was my reason for buying a PS1 in the first place, back in 1997.
IX is a lot more light-hearted than some of its predecessors, and also fun to play. It felt a little shorter than VII and VIII to me, but I'm sure it wasn't - it's still 3 or 4 discs long. It also has Vivi the black mage in, who is possibly the most adorable thing ever.
VIII is one that seems to divide people, with a lot of gamers citing it as being way down on their lists, Squall being 'too emo' and a whole load of other things. I enjoyed it once I got my head round the fact that it wasn't the same as VII. I still don't like the Draw system though. Spoony has the right of that - it can get very tedious. This is why I'd recommend trying it after some of the others - you might be more likely to forgive it some things and persevere.
It's a good story if you do.
X and X-2 should probably be played sequentially. X is certainly quite linear, but then if you consider the story of Yuna's pilgramage, it's understandable so I'm not to bothered by it. X-2 uses the same world but a lot more open. Tidus... Hmm. He's not one of my favourite protagonists (that honour belongs to Vincent Valentine - man, Turk, God), but he'll see you through. Just try to close your ears through the infamous laughing scene.
As for the rest... I'm actually one of those weirdos who found XII a lot of fun, at least once I was able to move Vaan out of the party and pop Balthier in there instead. And the early games of I to V give you a great insight into how the series has progressed. The only thing I can't comment on yet is XIII, since I haven't really started that yet (too many other games, work, blah blah blah).
In all honesty, play the first one you can get your hands on (aside from XIII I think) and see how you get on with it. Don't be put off by old graphics or sprites. They aren't perfect games, don't expect them to be. But let them tell you their stories and Uematsu's music take you along for the ride.
