What a question to ask... Prepare yourselves for eleventy million different recommendations and suggestions. Because this is the thing about Final Fantasy series fans - everyone has a different favourite.
Ultimately it boils down to your own personal tastes as a gamer. Does playing in twenty year old graphics bother you? Admittedly some of the older games have had a polish up, so you can get IV for the Advance and PSP for example, or even III on your iPad. But if you have a go at VI, then you'll be using the original sprites, and VII's combination of 2 and 3D is a bit dated now. Personally I don't really give a monkeys what it looks like so much - that's not why I play RPGs.
Personally, I would suggest one of either VII, IX or X.
VII is the poster child for the series. There are people living in caves who'd be able to recognise a silhouette of Cloud and his Buster Sword. It's not overhyped. It's not overrated. This is not to say it's perfect. There are bugs, some of the translations are just bonkers, and there are plot holes. It just might not be your thing. You might take one look at the blocky mange characters and laugh till your sides split (you may laugh now; in '97 we were bowled over by how awesome this thing looked). But it got me hooked on the series back in 1997 enough that I've played every other game in the franchise, which isn't a terrible recommendation, is it?
IX is probably the sweetest installment I've played, strange as it sounds. It has a more lighthearted and humourous vibe than the three preceeding entires. The story is still epic in scope, and the environments are beautiful. There's a lot of references back to the original games as well in here, which you start to appreciate the more FF's you play. It also has Vivi, which is reason enough alone to put the disc into your PlayStation.
And then there is X. Since this is a PS2 release, the graphics are a bit tighter, closer to what we expect these days. It's also the first one to have voiced characters, which makes a bit of a difference. It's more linear than the previous games (though that makes sense for the story, don't worry), so you shouldn't end up looking at a map going "Well, WTF do I do now?". Battle system is good, too.
The alternative is to start at the very beginning, working your way through. That'll give you the best idea of how far the series has progressed, but might also make your brain explode. The very early ones are... unforgiving... in places.
The thing to remember is that the games aren't clones of each other, they vary wildly in many ways. Some of the things the developer tried were brilliant. Others are best consigned to the bins of history. The trick is to take each as it comes, and enjoy it for what it is.
Enjoy...
