Let's see!
VI is my personal favorite, and has one of the best character-oriented stories of the entire series. It is also the game that goes deepest into the theme of personal tragedy and loss, and overcoming that.
However, being from the early 90's, it suffers from limitations like visual medium (sprites, negligible though) and writing (very painfully silly in some places). Despite these, the game can and does leave many people crying more than once. And even though the lead villain is not nearly so developed as all those that came after him, he's quite a trip (and the closest we get to a justification of his actions is actually a bit heartwrenching, even if it only comes at the end). The paradigm shift in playstyle that comes after a certain point in the game (trying to avoid spoilers) is pretty jarring though, and makes a guide even more necessary than normal if you want to properly complete the game or even get ANYWHERE, depending on individual obtuseness.
VII is HUGELY overrated. Hugely, insanely, to a painfully idiotic degree. This, however, does NOT mean it isn't good. Or even great. Move aside the rampant jackass fans and you find that many if not most people who play the game really enjoy it. The characters are again well-fleshed out, the story is very good (though not nearly as complex as some people think), and the villain is actually tragic, which is the first time in the series this occurs (Golbez doesn't count). In-game systems are easy to navigate, and the obscenely hard minigame silliness that crept in later in the series is mostly absent here. VII is a solid, good choice to play for a Final Fantasy game; it hits almost everything right.
VIII Is a game that focuses primarily on the characters over the story (which, thanks to the time-related elements involved, is a bit of a mindf***). It has also received some flak for awkward in-game systems (junctioning, the lack of accessories). For this one, people seem to either love it or hate it. No in between. As I also know little about it beyond this, I apologize for the short blurb.
IX is said to be the perfect Final Fantasy game by the CREATOR of the whole series. The characters are good and developed, the plot is complex without being stupidly-so, and the lead villain (despite looking ridiculous) is the most powerful and effective throughout the series (with the possible exception of Exdeath from the last ten minutes of V, regarding potential power anyway), while simultaneously also being tragic in his own way. As a bonus, IX contains tons of shoutouts and references to the whole rest of the series, which makes it rather ideally suited to what your situation is.
The CG scenes are VERY ridiculous looking thanks to the style they chose. However, they're so sweepingly epic at times that you completely forget this. IX is easily one of the best FFs you could play without knowing about the series, and remains so even if you DO.
X has one of the best overall stories of the series. Unfortunately, if you have trouble with things like suspension of disbelief, or thinking in any way besides "linear", you won't appreciate it. It's very pretty both in-game and in cinema scenes, but the voice acting comes across as stilted and awkward fairly often thanks to mistakes made in the direction (it's very hard to speak in English while matching lip flap made for the Japanese language). The level up system is also VERY needlessly complex and non-linear, as it was created to get AWAY from the standard level up route that JRPGs in general take. The characters are, once again, well-developed (Yuna is one of my absolute favorite archetypes in fiction, but that's just a personal thing). The actual battle controls and system are great and fun. The antagonists are well-developed, in some cases to such a degree that you realize this is around the time in the series where they started making a real effort to create more elaborate, "real" feeling characters. And the ending is...well, I won't spoil. But it's worth it.
XI is an MMO. Meaning it carries all the advantages and all the problems those games bring with them. The in-game visuals are limited thanks to this, and combat looks rather awkward.You NEED a party with you to do almost anything after a certain level, and "soloing" effectively is hard unless you know the vagaries of the system. The story is apparently pretty good, but in all honesty you should give this one a miss if you're looking for a single good example of the series to play for fun.
XII has the single most well-developed world in the entirety of the Final Fantasy series. Not necessarily characters or plot, but the WORLD in which all these things occur is so wonderfully realized that it can draw you right in without you realizing it. XII bears similarities to VI in that there isn't a single main character: the plot follows all six protagonists, and each has their own reasons for being there and doing what they do (some better than others). The art style is jarring to some (Lots of pastels), and the battle system is a significant departure from almost every game before it (using, as it does, a variant of the system from XI). Antagonists (and characters overall) are again quite well-realized and developed. Not quite as 'real-feeling' as X *perhaps*, but still interesting. And they have to be, since the story is driven by them. The overall plot is MUCH smaller-scale than a typical Final Fantasy game. Not a bad little game to pick up and play (got perfect ratings in Japan), but not *quite* a pinnacle example of the series.
XIII is...weird. Square-Enix went out of their way to do EVERYTHING different with this game. The standard turn-based battle system is swapped out for a real-time one. You only control a single character on top of this, and if that character dies you lose, making the game a kind of prettier, more strategy-intensive version of Kingdom Hearts. The standard Final Fantasy classes are gone too, melded into and replaced with a whole new list.The game is ridiculously pretty, to the point you can literally stop in places and just staaaaare at scenery. Again, it's the characters that drive the story: caring about them is what makes you keep playing. However this means if you're easily annoyed by them (a complaint of many people who played the game) then you're going to have no reason to keep playing at all. It's another love-it-or-hate it situation.
I personally adore the characters in XIII, and am tempted to play it for that reason. However the limited player interaction/freedom and the sheer awkwardness of everything new Square Enix tried to do with the game means you can safely give it a miss.
XIV was so bad upon its initial release that Square Enix gave everybody playing it a month's free play. Again, it is an MMO. Wait a little bit before trying this one out.
Given the options in the poll, I recommend IX or VII. But no matter which game in the series you choose to play, GET A GUIDE. Gamefaqs will do fine in most cases. But I'm not kidding when I say you'll NEED one.
I didn't include anything before VI because the list started at VII, and I was indulging a bit of bias by including my favorite in the series. There are strong arguments to be made for all the games before it however, including one of the most awesome last stands by any character in the series (V), one of the most sympathetic heroes and well-made (if a bit silly at times) story (IV), the introduction of and delightful variety of the job system (III), and games that are just fun in a very simple way (II and I).
Phew. Hope that helped.