I don't know anything about category theory, but asides about it occur so often in my daily reading that I really should; moreover, since I frequently see it applied to formal syntax and network descriptions, two areas I am interested in, I am probably therefore missing some important tools from my toolbox if I don't look it up.
To Read

The traditional vector for this stuff in these dotcom times: category theory for functional programming
 Mazzola, G. (2002). The Topos of Music.
 The catster's lecture series, indexed
 Chris Aldrich's UCLA Category Theory Summer Study Group
 Emily Riehl: Category theory in context is the free version of a book coming out soon, recommended to accompany Baez' newest course
 May as well file it here: Jerermy Kun's Algebraic Topology series

John Baez has a wealth of, to my innocent mind, provocative uses of this category thingy
 Diagrams
 Elevator pitch (Possibly an elevator to the core of the earth)
 Networks (Once again, a tome amazingly general take on networks, although perhaps that should not be surprising given the generality of the tool)
 a tale of nCategories

Maarten Fokkinga, A Gentle Introduction to Category Theory  the calculational approach

Barr and Wells
 Toposes, Triples and Theories
 Category Theory for Computing Science which is a highly recommended textbook with a cheap cover price, but only sold through a tedious shipping process at extortionate rates by the university of Montreal. (Don't bother looking on Abebooks, it's about $US130 there too. I imagine a thriving bootleg market for this one.)

Abstract and Concrete Categories : The Joy of Cats is a reprint online textbook by by Jiří Adámek, Horst Herrlich, George E. Strecker

David Spivak's Category Theory for Scientists/the Sciences

Bartosz Milewski's Category Theory for Programmers
 or Phillip Wadler, Category theory for the working hacker
[…] on why category theory is relevant for developers, discussing the principle of Propositions as Types connecting propositions and proofs in logic, and types and programs in computing.
 Tom Leinster's Category theory textbook
 Steve Awodey's Category theory textbook
 Everyone's secret alma mater, good old Wikiversity, has an Introduction to Category Theory
 on a similar bent, Stanford's Encyclopedia of philosophy article looks good and has a killer bibliography
 Alexander Kurz's listing
 The inevitable Math overflow question
 Andrea Asperti and Giuseppe Longo. Categories, Types and Structures. Category Theory for the working computer scientist
 Eugenia Cheng’s Youtube lecture channel
 John D Cook, Visualizing category concept dependencies_