VII: Started playing this when I was about 8 years old. Me and my friends shared a copy that was borrowed from another friend, and I ended up not beating it for a couple years. I'd just get lost for a while and not really even care because I was just having so much fun playing it, and pouring over GameFAQs guides and such. It was the first RPG I ever played, and to this day it's pretty much my favorite game just because of the massive amount of nostalgia that is attached to it.
VIII: The extent to which I was excited for this cannot be overstated. I remember my dad paying me $100 to de-weed the front lawn (he was being nice giving me that much money, but I pretty much earned it because that shit took forever), and finishing a few days after the game came out. I immediately begged my mom to take me to EB to buy it. It was the first time I ever bought a new game for $50, as I don't think I had ever really even had that much money at once ever before. We got back from the mall kind of late so I only got to play it for about an hour before I had to go to bed (I was in 4th or 5th grade). Although I had played the shit out of VII, I still was really young and didn't really know much about the series as a whole, and so I kind of expected it to be a lot like VII except maybe with a different story. I'm really not sure what I expected, but I remember being kind of... not disappointed, but certainly taken aback, as I was walking through a university-like place talking to people and not fighting anything. I wasn't crazy about the character designs, either. I remember being kind of upset that I didn't get to actually fight anyone that first night of playing because I had to go to sleep before I was finished running around the Garden. I warmed up to the game though, and eventually beat it, but it wasn't the game I wanted it to be when I bought it.
IX: Skipped this one for some reason, didn't end up playing it until after I played X. No real meaningful memories attached to this one.
X: My friend's brother stole a PS2 from some department store when it came out (took a display model into the bathroom, slipped it into his backpack) and this was one of the games he bought for it. I actually just watched him play it for quite some time, marveling at the graphics. I eventually got a PS2 of my own and got this game pretty quickly. Not my favorite in the series, pretty good though.
After that I kind of lost interest in the series. X-2 was had a great battle system, and was pretty fun actually, but in a very different way than the classic entries. I got pretty far in XII, but I still haven't bothered finishing it. I got used to the gambit system but I never really learned to love it. Haven't played XIII at all. Replaying the older games in the series the last few years has been fun, though. So far I've beaten I, IV and VI. I'd say IV, VI, VII are my favorites, VIII and IX are great as well, X was very good, XII was okay.