I was introduced to the series by a friend. I distinctly remember us sitting around playing Nintendo DS games, and how I'd brought my Action Replay. My friend asked to borrow it for this game called Kingdom Hearts. I let her borrow the AR, and she let me watch her play the game over her shoulder. Funny thing was that the AR somehow corrupted her save file, so she had to start the whole game over. I felt terrible, but she took it really well- something about how she'd been meaning to replay it anyway, I think. So I got to watch the opening parts of the game and was fascinated- everything looked pretty cool, and I'd always liked fantasy stories.
It was a couple years after that when I was deciding what I wanted for Christmas when I thought of that game my friend had been playing, and so I decided to ask for it. Not remembering the title completely, I wrote "Kingdom Hearts 365 1/2 Days" on my list. I ended up getting the right game anyway, because, as I was told, "It was the only one with numbers." I spent about a week playing the story mode, got bored of Roxas, and... promptly switched to mission mode to find a character I liked better. It was a couple years later that I learned the Organization were actually the villains and that the character I ended up liking was destined to die, so... that's when I decided to give Dream Drop Distance a try. Sora annoyed me, but Riku seemed okay, so I went along with it, was able to get through the game for the most part and have a good time doing so. DDD gave nice summaries for the other games, so I was finally able to know what was going on (sort of).
I recently got a PS3, so now I can play the final mix versions! But I only have the collection that includes KH2, BBS, and the Coded movie. Somehow, though, my patchy experience with the series was enough to keep me interested- I ended up watching game footage and reading a lot of summaries online- and I'm still following this thing... It's more about the lore and the world than anything else, since I don't care much for the main characters or Disney in general, and have no experience whatsoever with Final Fantasy. I guess it's more of the idea of the series that keeps me engaged- all of these different elements in a wild story that could probably go on forever. It's great material for role-playing, which I do a lot of with another friend of mine.