My thoughts are similar to what's been said above, but I'd like to emphasise one suggestion which perhaps is only implicit in other people's answers (or which they might disagree with).
For me, learning category theory was done only after I'd learned a little bit about several different things, or at least struggled through courses on such. Thus for me its main attractions are twofold:
It often provides a unifying perspective on constructions which live in different subject areas but which seem awfully similar: e.g. the first isomorphism theorem for groups, rings, Banach spaces (suitably interpreted), and so on. Pushouts and pullbacks crop up all over the place too, once you get accustomed to the abstract picture. And Galois connections are somehow the "right" way to think about several condition-laden duality theorems, in both analysis and algebra. (Since you mentioned representation theory, at a very basic level there's the example of a group as a 1-object category - in which case representations are just functors from this category to Vect, intertwining maps between representations are natural transformations between these functors, and the restriction-induction duality is an adjunction between appropriate categories.)
Sometimes, if one wants to prove an identity or construct a map (homomorphism, continuous map, whatever), one direction or part turns out to follow "just from formal reasoning" or "just by abstract nonsense". So having some category-theoretic language at one's fingertips can be a good way to quickly work out which bits of a problem depend on the particular structures at hand. Again, various arguments about reflexivity and (weak) complementation for general Banach spaces can IMHO be clarified by knowing that the embedding of a Banach space in its bidual is a special case of the unit of an adjunction.
Thus, while I generally agree that it's better to learn category theory in the context of your particular discipline, I would say that it's also a good excuse to learn about parts of other disciplines; and conversely, the more examples you have from different parts of mathematics, the easier it'll be to understand the category-theoretic notions they exemplify.
Oh, and I quite like CWM, but I had to read it in conjunction with some course notes. So I'm not sure what it's like as a stand-alone source.