To go through a few quick recommendations, from most to least...
Final Fantasy IX
If there ever was a "definitive" game in the FF series, I'd have to name FF9; it's the nostalgia-driven homage to all the games which came before it, taking the best elements of each and bringing it all together as wonderful experience from start to finish. The storyline is relatively easy to understand, but VERY well-paced and enjoyable. The characters all have their own development, and are quite endearing in their own right. While not the best soundtrack in the series (in my opinion), it's still one of the better ones. You can't go wrong with this game, period.
Final Fantasy VI
FF6 is the first game which departed from pure fantasy and started incorporating some science fiction elements into the game, but it's best known for having some of the best characters in the series and quite possibly the single best soundtrack in all of gaming. The legendary opera scene, with the renowned Aria de Mezzo Carattere, is one of those things which remains to be topped... it's been 20 years at this point, it's that good. The game itself is quite good as well, but the focal points here are definitely the storytelling, characters, and music.
Final Fantasy X
FFX is the first of the "modern" FFs, and takes place before Square started reworking how their battle system works; it still closely follows the old ATB system (which all my other recommendations have), just changes the active timer for a pseudo-cooldown approach. First FF game to feature voice acting.
Final Fantasy IV
Now, for the "classic" game of the series. FF4 may not be the best in the series, but it more or less established it; the first game to feature a compelling storyline, and it also introduced the ATB system. It may be a bit dated at this point, but it still works quite well.
Final Fantasy VII
The praise of this particular game is often overblown in my opinion, though one can't deny that how successful it was at popularizing the series nor the fact that it's still a great game in its own right. I'm not a big fan of it myself, but it's still worth playing through once.
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The first picks from me would be FF6 and FF9.
FF6 is one of the best classic games out there, though it may be a bit dated for some. FF9 is just modern enough to overcome that issue, and is potentially the best overall package in the series; even if it lacks the stand-out moments other games in the series have, it's has some of the best balance throughout.