I liked XIII a lot, personally. You have to be okay with some Japanese cultural differences (mainly their mannerisms, characterizations, and the fact that they like monologues in their stories), but overall it's a fantastic game. The battle system is my favorite in the series so far, and though the overall story isn't quite up to par with some of the other Final Fantasy games, the incredible character development and inter-relational drama of the first 20 hours more than makes up for that. It's not quite my favorite Final Fantasy game, but it's up there. One of my favorite RPGs of the millennium so far.
EDIT: I will add that I think VI, VII, and X are the best so far.
If you want a truly incredible example of storytelling with the limited technology of the Super Nintendo, play that one. A truly amazing game with genuinely emotional storytelling and one of the best soundtracks to grace a video game. Gameplay is good too, of course, but that was never what made Final Fantasy amazing: the story, characters, and all-around artistic quality is overwhelmingly good, so play it.
VII was the first to go 3D (still an old-school version of 3D, but nonetheless different from the old-school 2D RPGs). Boasting another incredible soundtrack, a long, engaging story, and more lovable characters, it's quite possibly the greatest interactive fantasy story ever told. The battle system is standard turn-based fare, but the customization outside of battle (namely the ingenious Materia system) is exceptional.
X is more up to par graphically, being made more recently, and it excels in many areas. Despite one particularly cheesy scene and patchy early PS2 voice acting, it is a fantastic game with a great character-driven story. The soundtrack is not the best, but still rather good (not to mention it has To Zanarkand, possibly the most beautiful piano song I have ever heard). It also has my second-favorite battle system of the series.
EDIT #2: For the record, Final Fantasy VII was the first RPG I ever played, and I quickly fell in love with JRPGs after playing that masterpiece of a game. Just throwin' that out there.