In answer to Andrew's question, I think it really depends on the student. I began learning category theory in my late teens because of the sorts of questions I was asking myself which, I discovered, could be answered through category theory. It just really "clicked" for me, and provided me with tools that I use every day in my mathematical life.
Sometimes I would find an application of category theory to an area I didn't know too much about, but because the application seemed pretty cool, I would be motivated to learn more about the area. My own feeling is that the category theory helped me learn mathematics more quickly than I otherwise might, in part because it helped provide broad conceptual frameworks in which to fit newly acquired knowledge. So in that respect, I am happy that I began learning category theory early on.
But category theory doesn't come naturally to a lot of people (some of the people who have answered or commented above, including some very distinguished mathematicians, don't strike me as having a whole lot of feeling for the subject). That's fine. If category theory does not come naturally to you, then simply learn category theory on a need-to-know basis, and try not to make up your mind what the subject is about (e.g., "doing away with elements") in advance. My advice is: don't force yourself to learn it unless you have a need to know (and my guess is you probably will, in tandem with other subjects).
Over time, while studying something that you've really latched on to, you may find some categorical reasoning coming into play, and marvel at how clean and efficient it is, and how it clears away conceptual clutter. Then you may be in a proper frame of mind to make a deeper study of what makes some aspect of category theory "tick", with some heightened appreciation of what category theory is good for, or how it can serve your ends.