In contrast to what some of the other answers seem to be saying, I believe that size issues play a very important role in category theory. Consider, for instance, the notion of complete category, i.e. a category having all small limits. Most "naturally-ocurring" categories, such as sets, groups, categories, etc. are complete (and cocomplete), and the ability to construct small limits and colimits is extremely important. However, these are all large categories, and a classic proof due to Freyd shows that in fact any small complete category must be a preorder (i.e. any two parallel arrows are equal). Thus, one of the most basic notions of category theory (completeness) becomes trivial if you aren't careful with size distinctions.
I also feel that more mathematicians should pay attention to set-theoretic issues, especially in category theory, and I wrote an unfortunately lengthy note myself on the subject, akin to Murfet's and Easwaran's pages linked to in Greg's answer.
However, for purposes of learning category theory, I don't think one should pay too much attention to any of this stuff. I think all you need to know, beyond naive set theory, is that some collections are "too big to be sets" (like the collection of all sets) but we can still manipulate them more or less as if they were sets, and we call them "classes." NBG and MK formalize this nicely with the "Limitation of Size" axiom: a class is a set if and only if it is not bijective with (i.e. "is not as big as") the class of all sets.
In relatively mundane, but intensely useful and practical, ways, the naive-category-theory attitude to characterize things by their interactions with other things, rather than to construct (without letting on what the goal is until after a sequence of mysterious lemmas), is enormously useful to me.
E.g., it was a revelation, by now many years ago, to see that the topology on the space of test functions was a colimit (of Frechet spaces). Of course, L. Schwartz already worked in those terms, but, even nowadays, few "introductory functional analysis" books mention such a thing. I was baffled for some time by Rudin's "definition" of the topology on test functions, until it gradually dawned on me that he was constructing a thing which he would gradually prove was the colimit, but, sadly, without every quite admitting this. It is easy to imagine that it was his, and many others', opinion that "categorical notions" were the special purview of algebraic topologists or algebraic geometers, rather than being broadly helpful.
Similarly, in situations where a topological vector space is, in truth, a colimit of finite-dimensional ones, it is distressingly-often said that this colimit "has no topology", or "has the discrete topology", ... and thus that we'll ignore the topology. What is true is that it has a unique topology (since finite-dimensional vector spaces over complete non-discrete division rings such as $\mathbb R$ or $\mathbb C$ do, and the colimit is unique, at least if we stay in a category of locally convex tvs's). Also, every linear functional on it is continuous (!). But it certainly is not discrete, because then scalar multiplication wouldn't be continuous, for one thing. But, despite the prevalence of needlessly inaccurate comments on the topology, the fact that all linear maps from it to any other tvs are continuous mostly lets people "get by" regardless.
Spaces with topologies given by collections of semi-norms are (projective/filtered) limits of Banach spaces. Doctrinaire functional analysts seem not to say this, but it very nicely organizes several aspects of that situation. An important tangible example is smooth functions on an interval $[a,b]$, which is the limit of the Banach spaces $C^k[a,b]$. Sobolev imbedding shows that the (positively-indexed) $L^2$ Sobolev spaces $H^s[a,b]$ are {\it cofinal} with the $C^k$'s, so have the same limit: $H^\infty[a,b]\approx C^\infty[a,b]$, and such.
All very mundane, but clarifying.
[Edit:] Partly in response to @Yemon Choi's comments... perhaps nowadays "functional analysts" no longer neglect practical categorical notions, but certainly Rudin and Dunford-Schwartz's "classics" did so. I realize in hindsight that this might have been some "anti-Bourbachiste" reaction. Peter Lax's otherwise very useful relatively recent book does not use any categorical notions. Certainly Riesz-Nagy did not. Eli Stein and co-authors's various books on harmonic analysis didn't speak in any such terms. All this despite L. Schwartz and Grothendieck's publications using such language in the early 1950s. Yosida? Hormander?
I do have a copy of Helemskii's book, and it is striking, by comparison, in its use of categorical notions. Perhaps a little too formally-categorical for my taste, but this isn't a book review. :)
I've tried to incorporate a characterize-rather-than-construct attitude in my functional analysis notes, and modular forms notes, Lie theory notes, and in my algebra notes, too. Oddly, though, even in the latter case (with "category theory" somehow traditionally pigeon-holed as "algebra") describing an "indeterminate" $x$ in a polynomial ring $k[x]$ as being just a part of the description of a "free algebra in one generator" is typically viewed (by students) as a needless extravagance. This despite my attempt to debunk fuzzier notions of "indeterminate" or "variable". The purported partitioning-up of mathematics into "algebra" and "analysis" and "geometry" and "foundations" seems to have an unfortunate appeal to beginners, perhaps as balm to feelings of inadequacy, by offering an excuse for ignorance or limitations?
To be fair (!?!), we might suppose that some tastes genuinely prefer what "we" would perceive as clunky, irrelevant-detail-laden descriptions, and, reciprocally, might describe "our" viewpoint as having lost contact with concrete details (even though I'd disagree).
Maybe it's not all completely rational. :)
Best Answer
There is a majestic paper by Mac Lane
whose opening line is one of the most beautiful I've ever read:
I've often wondered what does remains of those suggestions, and I strongly recommend you (if you haven't already) to have a look at this inspiring note: it is a masterpiece of neat exposition and it is replete with the hope that category theory becomes deeper and stronger, with the passing of time.
It is organized in brief, lapidary short sections, and proposes several directions in which category theory can, should or will go: in a few words
I'll leave you the pleasure of reading the note for yourself. My opinion (which is only the humble feeling of a young craftman) is that there are few items we can feel we have completely solved, even 50 years later.
Of course, today we have more higher category theory than we could ever hope for. Of course, we have a few people working in axiomatic cohesion. Of course, we have people in type theory and in HoTT. And also, we are lucky because today few people question the "importance of being abstract". But there is so much still to do!
And the best way I can explain what I'm saying is by adding an item to the otherwise complete Mac Lane's list.
Category theory is huge, but few people outside pure mathematics apply it. Many people know that it exists, but few people appreciate its elegant, tautological statements and try to apply it to different things (those who do it are outstanding mathematicians, way better than I will ever be). This is what makes pure mathematics vital: a bunch of flippant engineers and physicists and biologists shaking it, breaking it, deforming it. We shall give other people tools to package immensely deep ideas in an extremely low volume ("rings are spaces"; "homotopy theory is localization"; "the Yoneda lemma"...).
Last, but not least, I feel we shall communicate why we feel lucky: category theory is an island of beauty in the already beautiful land of mathematics, and we are in love with it. We shall communicate the bliss we feel when we do it.