To complement the other answer and comments: the utility of ideas of category theory is not at all limited to "algebra", either. For example, the product topology on an infinite product of topological spaces had always struck me as disappointingly weak, and I wondered why that was "the definition". In fact, that construction is a construction of (a model for) the categorical product in the category of topological spaces. That is, instead of just being a definition we've inherited, it has functional properties. In fact, with this bit of hindsight, it seems to me perverse to "define" so many things without explaining what is supposed to be happening. A categorical characterization is often much more informative.
Another example, algebraic, but not "fancy": what is an "indeterminate", after all? A "variable"? Certainly there are heuristics that we'd tell beginners, and we know how to use "indeterminates x,y...", but what are they? One precise form is to say that $\mathbb Z[x]$ is the free ring with identity on one generator $x$... meaning that, given $r\in R$ in an arbitrary ring $R$ with identity, there is a unique ring homomorhism $\mathbb Z[x]$ to $R$ sending $x$ to $r$.
In a quite different direction, the topology on the space of test functions on $\mathbb R^n$, or even just on compactly-supported continuous functions, is a colimit. For continuous compactly-supported, it is the colimit of Banach spaces $C^o_K$ of continuous functions on $\mathbb R^n$ supported on compact $K$. In contrast, the "definition" given in Rudin's "Functional Analysis" for the test function topology is actually a construction, and the following section proves several mysterious lemmas which, I only realized later, were, in effect, verification of the colimit properties. (Indeed, Schwartz overtly used the notion of colimit c. 1950, but use of such notions had been out-of-style for U.S. analysts for many decades. Some of that may be anti-Bourbaki reaction, even though Bourbaki did not use category-theory ideas, either.)
In the short term, it is usually possible to "get along" without overt use of category-theory terms or ideas. However, the more things one finds reason to remember, the more imperative there is to organize them well, to eliminate redundancies and duplications and waste, etc. Category theoretic ideas are very helpful in this regard.
(This is not to say that "formal" category theory is necessarily as broadly useful, in the same way that, while set theory is undeniably useful, the utility of a highly developed formal or axiomatic set theory is probably not nearly as useful as the basic parts.)
I would really recommend that you plunge in and seriously read Aluffi's Algebra - chapter 0. The book will cover many many topics in abstract algebra including group theory, ring theory, field theory, as well as more advanced material like homological algebra.
The book is marvelously written which is a reason on its own for reading it. But, since you are looking for the category theory perspective this book is really what you are looking for. It does not assume any category theory, but instead develops parts of at as you go along, exemplifying everything with the algebra being developed at the same time.
Depending on you level of comfort with abstract ideas, you might find that you want to reinforce reading the book with reading a more elementary text on group theory. Rotman's Group Theory is excellent.
I really don't like Fraleigh's book, though I know it's popular. In my opinion the order in which things are presented makes little categorical sense.
The nice thing about Aluffi's book is that when you finish it you can truly say that you know the chapter 0 of modern algebra. It really gives you a very sound foundation of all of modern algebra.
Best Answer
You can get a fair amount of category theory in the masters of parisian universities, mainly oriented towards algebraic geometry and algebraic topology. Also, you get a fair amount of categorical seminar around Paris (go figure...).
It depends on the year you attend, but Université Paris 6 (Pierre et Marie Curie) usually have courses on étale cohomology and sheaf theory for algebraic groups. Université Paris 7 (Diderot) usually have courses about homology/homotopy with a categorical point of view. Université Paris 13 (Saint-Denis) offers courses about homotopy and operad theory. Also, most courses can be followed independently of the university you registered in.
A little closer to Italy, there is quite a categorical team in Nice (Université Sophia-Antipolis), mostly around Carlos Simpson and Clemens Berger I would say. Definitely a good place for a categorical PhD, however I'm no sure it reflects into the master there.
Just my two french cents !