I remember the first thing I learned which sold me on category theory: the arrow-theoretic definition of injections and surjections. Thinking about how to define an injection without being allowed to say "element" is a good, and possibly fun, exercise. 

Besides this, +1 on Qiaochu's comment - realizing what "coproduct" means in the category of groups vs the category of sets is another great "aha!" moment.