In KH1, the part where Riku shows up in Monstro. I hadn't really expected Kingdom Hearts to have much of a story before that point, and that was the bit that made me realize that there was something more there.
Also in KH1, the scene where Sora plunges the Keyblade of people's hearts into his chest to free Kairi's heart. By that point I was pretty on board with the story, but I still was absolutely not expecting that. In fact, I'm still a little surprised they'd do something like that.
In KH2, the end of the prologue with Roxas. "Looks like my summer vacation is over" is a very memorable line, but honestly the whole final day was great. Especially since Roxas really clicked with me as a character, and I was pretty much ready to spend all of KH2 with him.