My favorite parts of each game:

I: Beating those stupid wizards in the marsh cave, which meant that the game was past the annoying 'not leveled up so you suck' stage.
II: Josef's sacrifice. Sad but heroic.
IV: The first fight with Rubicant. The greatest villians are the most polite.
V: When Gilgamesh tries to use the 'ultimate' weapons and meets with little success.
VI: A lot of video game bosses try to destroy the world, but Kefka actually succeeded! That was so awesome.
VII: Temple of Ancients. For some reason it's my favorite dungeon ever. Maybe it's the music, or the fact that it's the last main area you get to play with Aeris.
VIII: The assassination attempt on Edea was one of the greatest cinematic sequences ever.
IX: Playing as Steiner and Beatrix to defend Alexandria.
X: At the very beginning Auron gives Tidus a sword and the kid struggles to pick it up, yet right after the battle he tosses it up in the air like it's nothing.
X-2: Running around in that moogle suit was strangely entertaining.
Tactics: Zalbag realizes Dycedarg is evil and joins you to fight him.