Final Fantasy IV up though XII (excluding XI), you can't really go wrong. Your first will shape your expectations, but keep an open mind and you should have little trouble moving forward or backwards through the series.
My personal favorite is Final Fantasy IX, as it's essentially a SNES-era Final Fantasy made with (then) modern hardware. Wonderful art direction and OST, tons of humor and probably the best, most well told story of the PSone games (not counting Tactics).
Final Fantasy VIII was also great, with believable world design and an interesting blend of traditional fantasy, modern and futuristic settings. Junction was broken and the Draw system annoying until you learn to Refine magic from items/cards, at which point it becomes a non-issue. The story is convoluted, yet still enjoyable, and the cast was pretty decent (especially Seifer Almasy).
Final Fantasy VI, despite not being my absolute favorite, is probably the best game overall. Final Fantasy VII was another solid entry, but not quite as good as that which came directly before and after it.
Final Fantasy X was great, as was XII, which is the perfect example of Final Fantasy evolution done right. Final Fantasy XIII is an alright game in its own right, but doesn't do much that makes it a good Final Fantasy. It's linear like Final Fantasy X, but without the excellent combat system and interesting cast/story. That's not quite fair; It seems as though it had an interesting story, but it was told so poorly that it could be hard to tell. Although, you may like it if Final Fantasy XIV's CG is what attracted you to the series.
Just jump in anywhere, basically.