TITLE: Category Theory usage in Algebraic Topology
QUESTION [24 upvotes]: First my question:

How much category theory should someone studying algebraic topology generally know?

Motivation: I am taking my first graduate course in algebraic topology next semester, and, up to this point, I have never taken the time to learn any category theory. I've read that category theory helps one to understand the underlying structure of the subject and that it was developed by those studying algebraic topology. Since I do not know the exact content which will be covered in this course, I am trying to find out what amount of category theory someone studying algebraic topology should generally know.
My university has a very general outline for what the course could include, so, to narrow the question a bit, I will give the list of possible topics for the course. 
Possible Topics:

unstable homotopy theory 
spectra
bordism theory
cohomology of groups
localization
rational homotopy theory
differential topology
spectral sequences
K-theory
model categories

All in all, I am well overdue to learn the language of categories, so this question is really about how much category theory one needs in day to day life in the field. 
Update
I emailed the professor teaching the course and he said he hopes to cover the following (though maybe it is too much):

homotopy, homotopy equivalences, mapping cones, mapping cylinders
fibrations and cofibrations, and homotopy groups, and long exact homotopy sequences. 
classifing spaces of groups.
Freudenthal theorem, the Hurewicz and the Whitehead theorem.
Eilenberg-MacLane spaces and Postnikov towers. 
homology and cohomology theories defined by spectra.

REPLY [10 votes]: I agree with Paul Siegel's very nice answer, and would just like to add one thing that's a little too long for a comment.
Depending on what direction you take, algebraic topology can become practically synonymous with higher category theory.  This can come in multiple ways.  First, the category of topological spaces has spaces as its objects, continuous maps as its morphisms, homotopies as its 2-morphisms, homotopies between homotopies as its 3-morphisms, etc.  Stated perhaps a little too cavalierly, the point of model categories is essentially to set up a general framework for studying higher categories that may (or may not) look like that of spaces.  But then, higher categories themselves are also a lot like spaces.  In this analogy, functors are like continuous maps, natural transformations are like homotopies, etc.  (The fact that there are these two totally distinct ways that spaces and categories interact really blindsided me the first time I saw it.)
Anyways, the point is that if you pursue algebraic topology seriously, you may eventually have to get very friendly with category theory and be okay with using ridiculous and scary phrases like "homotopy left Kan extension" and such.  It seems that usage of higher category theory in algebraic topology is very much on the rise, so it's possible that in twenty years, algebraic topologists will have no choice but to become well-versed in all this stuff.  (I'm certainly not.  Not yet, at least.)  Just a heads-up.