I was 13 years old and my friend lended me... Tokyo Mew Mew. Before that I had been checking out Detective Conan in the library but never lended any since I wasn't that sure of what it was (and if it was appropriate to show to mom!).
The first one I bought was Death Note. I remember it well since I had to get it ordered from this tiny comic book store full of board games and whatnot. The 13 year old me was very weirded out! :D