Where do I learn basic category theory? Anything better than just perusing wikipedia?
also some preliminaries (books by Simmons, Awodey and Lawvere/Schanuel, spivak):
Only requires some comfort in proof writing, at least what would be covered in a first course in discrete math.
Having some Linear Algebra, Groups, Rings, Topology, provide some concrete structures to help contextualize the material but it is not required.
As an example, the University of Chicago, my alma mater, doesn't expose CS students to Category Theory. Of all the places it might be where you's expect it the most. It has a notoriously "theoretical" computer science department tied closely to the mathematics department and is where Category Theory was invented.
Even in a math degree, one wouldn't typically touch on Category Theory in a classroom setting until one studied Algebraic Topology. Before that students might come across it as a neat sideshow, but nothing they'd be interested in using to solve an actual mathematical problem. There are plenty of students who get a BS in mathematics without ever making a serious go at it.
It's also not really that useful as a first-order field of mathematics. It's mostly useful as a way of organizing other mathematical things and as I kind of general vernacular. Even mathematicians sometimes lovingly call it "abstract nonsense".
I see CT as a kind of mathematical interstate system. If you want to go back and forth between Chicago to LA it's fantastic, but most of your work is being done in Chicago or LA itself. The interstate system itself isn't that interesting most of the time.
More specific evidence to go with jfarmer's: I read mathematics at the University of Cambridge -- like UChicago, an absolutely first-rate institution and not one that gives its students an easy ride -- and there category theory is not part of the undergraduate curriculum. It is one of the courses you can take in "Part III", which is a one-year taught master's degree.
 Until very recently, the best way to describe it was "a one-year taught master's degree that inexplicably isn't actually a master's degree". But they've fixed that now.