It really depends on what sort of style you like in your games. Many of them have a fairy tale aspect [see many of the originals, and IX], but many also have sci-fi fantasy hybrid themes [VI, VII, VIII, X, X-2, etc.] and there's also the question of whether or not you like stories centered around the characters rather than the world, or the world controlling the characters, etc. Final Fantasy games are, for the most part, different from each other for different reasons, not just graphically, or even story based, but mechanics change, summons are different, team set ups, strategy, etc. Also, while I haven't played much of Tactics, I'm kind of surprised nobody's mentioned it... Like really?
I believe X is a great introduction to the series, along with IV, VI, VII, XII, and of course my absolute favorite, IX. IX is great because it encorporates all of the themes and references and molds it all into a lovable game with great characters and atmosphere, the soundtrack is [in my probably biased opinion,] the best.
But, regardless of my obsession of IX, X was my first FF game and honestly, I'm glad that I started with that one.
X is having its HD remake coming out soon, IV is on the DS and is pretty great but there are multiple versions too, VI idek I know its original is out there, VII is downloadable on PSN, XII is still on PS2 and I'm surprised there hasn't been an HD remake in the works for that considering but eh. IX is also on PSN.
This post is really disorganized and babbly but seriously IX is the best I'm out