The first one I ever played was 1. I didn't beat it (I didn't really understand the concept of level grinding at the time) but it was fascinating. It was nearly double my life later before I discovered the joys of non-console roleplaying.
I played a lot of the games with the Final Fantasy namesake, though I didn't beat many of them. I actually loved Mystic Quest, though I know I'm in the minority for it.
7 and 8 I played on the computer and enjoyed both, though my Disc 1 of FF8 is scratched. I tried to start playing it again a few months ago and it wouldn't work ;_; I remember how my computer was so crappy that I had to play 8 on a quarter-screen (1/4 the size) just so that it could handle the game.
9, 10 and 10-2 were rather meh. I don't really remember much of anything about the stories, and my favorite girl from 10 got knocked up and didn't get to go along for the sequel, instead replaced by Generic Goth Girl #19417.
I liked 12 in many ways, though I can see the argument that the main character was just along for the ride.
13 and 13-2 were pretty meh, and 13-2 (very minor spoiler) had one of the worst things that you could do to a game ending: not have a freaking game ending. To Be Continued? Really? Why do you hate me?
I want to play 6 someday, but I'm worried about how dated I know it will inevitably be. Since I just replayed Mystic Quest, though, maybe I can get over it.
What series? There are only four things that every Final Fantasy has had, everything else can be up for grabs in terms of plot and story and world. Those four things?
Plot Device Crystals.
Airships (Which may or may not have a Cid attached to it.)
The Boss, AKA Bahamut.
And finally, the most recognizable feature of the Final Fantasy franchise: the Chocobo.
Every game that doesn't have a -2 at the end of it has been a totally and completely different experience.
I actually like the way that they test out new things, keep some and drop others. The mechanics are a completely different issue, though I can certainly go into that.
But then, I'm crazy enough that I enjoyed FF Mystic Quest, played a lot of 4E D&D and am looking forward to 5E of it. Innovation is an exceptional way to find the best new ways to do things, even if some of them don't always work.
I can see the argument for having a more coherent setting, but it also means that every new game gives a completely new area to check out. Zelda's followed this example in many ways as well, but it keeps it from having the feel that it's the same game with a few minor splashes of paint.