I'll map out my feelings about each game from V to IX, so you can get a gauge of what they offer.
V
Last "difficult" FF game (most enemies kill you with ease, but they aren't insurmountable)
Last "classic" FF game (Warriors of Light, Crystals, Medieval setting, etc)
Customizable job system (same as the Dressphere system from X-2 that you may have heard of)
Decent characters (but limited development and lame villain kills them)
Hilarious recurring boss (Gilgamesh is the star of the show)
This is probably the best case if you'd like to experience the old-school games without the difficulty level of I-IV. Sadly, it can be unforgiving if you're not prepared. The characters and plot don't help either. Get it if the others interest you.
VI
First "modern" FF game
Tremendous character development (probably its best selling point)
Amazing Villain
Best music in the series (Don't take that lightly)
VI sets the bar to which every game in the series adheres to. Story, sound, music, character development; everything in this game is spectacular. It's also less challenging than its predecessor.
VII
Amazing Cinematics and Setting (brought down by AWFUL block models)
No "character classes" (Materia is a blank slate, and offers more flexibility, even compared to VI)
Tremendous "flow" (by that, I mean moving from point to point feels organic, dynamic, and seamless)
Deep storyline (convoluted is the right word, but not nearly as crazy as people make it out to be)
More "sci-fi" than VI (which felt more "steampunk")
Mature (This is the most mature FF game. Not just swearing, sex, etc...but the grimy, seedy, grown up asthetic adds to it)
I admit VI is the better game. That doesn't excuse VII from being the most "eventful" in the series. It has so much going for it that it is an experience; not just as a FF game, but as a cinematic achievement. It hasn't aged well, but the grown-up, ballsy asthetic shines bright. Given that a lot of games in the genre have subverted to "J-pop/Anime" or teenager cliches, VII stands stall as one of the truly "mature games" the genre has to offer. It took everything VI had and ramped it way up. Both VI and VII have to be played; no excuses.
VIII
Broken Gameplay (Abusable in every fashion)
Ages better than VII (graphics and sound especially)
Weak Romance (borderline goofy, nonsensical, illogical...any other word to describe absurd sums this up)
Villains feel ineffectual (moreso than Kefka or Sephiroth do)
Slower Pace (this story and setting definitely feels more like a chore to get through than VII does)
Still accessible, still fun, but good god, Disc 3 onwards is just awful. I don't want to spoil it. Just...take my word for it. If you hate romance, you will gouge out your eyes and pour acid in your ears.
IX
Callback to old-school FF
Character Classes Return
Harder than VI-VIII
Eccentric villain (has great style and execution, but he's seriously *** looking)
Dynamic main character (Cloud and Squall fit that "emo" image)
Romance subplot doesn't suck (not as abhorrent as VIII, not as effective as X)
IX is everything one would want in an old-school RPG in the modern era. It continued Square's ascent in quality (in every area). I don't think the game flows as well the modern FF styles, but is one of the best games you will find.
Start with VI or VII. Venture off to IX or V for a more classic approach. VIII is good, but that romance plot and growth system are grotesque.
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"Should I slice off your tongue or break your countless bones? Either way, I get paid; you'll bleed to death and die alone." - Blood Money