Since category theory is rarely taught in undergraduate or even graduate classes, I have occasionally been asked for the best way to learn about it. This page provides a summary of my answer.

Learning Category Theory

What follows is mostly a list of resources that I personally found useful for learning category theory.

The Catsters have uploaded videos of short lectures on various concepts in category theory.
A good starting point for seriously learning category theory is the first chapter of "Toposes, Triples, and Theories" by Barr and Wells (avaliable for download here). This chapter has a wealth of good exercises that relate the rather abstract notions of category theory to concrete examples.
Francis Borceux's three volume "Handbook of Categorical Algebra" gives a very detailed account of all the basic and many advanced topics in category theory. The detailed proofs make it a bit more accessible than the standard reference "Categories for the working mathematician" by Mac Lane.
The web column "This Weeks Finds" by John Baez often provides good intuition and pointers to the literature for some of the more advanced topics in category theory (and other parts of mathematics).
Australian category theory unfortunately has a reputation for being even harder to absorb than "ordinary" category theory. A good starting point for that would be Ross Street's book "Quantum groups," the "Review of the elements of 2-categories," LNM 420, by Kelly and Street, the 2-categories companion by Steve Lack, and Kelly's book Basic Concepts of Enriched Category Theory.
Finally: TAC and TAC reprints

Doing Category Theory

This is a list of a few sites everybody should (and probably does!) know about, here mostly for the sake of completeness.

The n-Category Cafe, a group blog on (higher) category theory.
The nLab, a wiki on (higher) category theory.
I wrote an alternate css file for the nLab, available as a "Stylish" style here
The nForum, its associated forum. Contributors of the nLab discuss and document their edits there.
Many category theorists can be found on MathOverflow.