Does category theory help understanding abstract algebra? I'm studying category theory now as a "scientific initiation" (a program in Brazil where you study some subjects not commonly seen by a undergrad), but as I've never studied abstract algebra before, so it's hard to understand most examples and to actually do most of the exercises. (I'm using Mac Lane's Categories for the Working Mathematician and Pareigis Categories and functors.)
To solve this, my advisor recommended me to get S. Lang's Algebra as a reference, but I don't know if that's the most appropriate book and if it's better to get Lang and study algebra through category theory or to study (with a different book and approach, maybe Fraleigh) algebra and then category theory.
PS: I'll have to study by myself (with my advisor's help), as I can't enroll in the abstract algebra course without arithmetic number theory.
 A: There is the book Algebra: Chapter 0 by Paolo Aluffi that might fit your needs.
It is a textbook on algebra (as the title says), but it uses the language of category theory from the beginning. Category theory is mostly used to motivate definitions using universality properties.
It is only in the last two chapters that the author introduces more advanced concepts from category theory (functors, abelian categories, etc.).
A: You can also give a try to other books on category theory which are more accessible than, say, the MacLane's classics.
Here they go:


*

*Conceptual mathematics: a first introduction to categories by Lawvere and Schanuel

*Arrows, structures, and functors: the categorical imperative
by Arbib and Manes

*Category Theory by Awodey
Lawvere & Schanuel require almost no math background at all, Arbib & Manes and Awodey are somewhat more advanced but should be at least partially available to a math undergraduate without much knowledge of abstract algebra.
A: I can recommend several much better sources that will ease your transition into both abstract algebra and category theory, Silva.
Lang is far too difficult for a first brush with abstract algebra, and MacLane is even MORE difficult for a neophyte in algebra. Category theory has VERY far-reaching conceptual implications for most of modern mathematics, not just algebra. So no, in principle, you don't have to learn abstract algebra to learn it, but that's where most mathematicians have infused it. This is because it's natural to organize types of structures into categories and that's really what algebra is all about: types of structures, i.e. sets with binary relations on them.
There are a legion of great abstract algebra texts, but my favorite is E. B. Vinberg's A COURSE IN ALGEBRA, available through the AMS. It takes a very concrete, geometric approach and builds an extraordinary amount of algebra from first principles all the way up to the elements of commutative algebra, Lie algebras and groups and multilinear algebra. It will help you learn a great deal of algebra very quickly and without the confusion of learning category theory simultaneously. Another geometrically flavored-but a bit more challenging-book is the classic ALGEBRA by Micheal Artin. Indeed, I think the 2 books compliment each other very nicely. Mastering both books will give you a very good working knowledge of algebra and you'll be more than ready to tackle Lang's book after that.
As for category theory, the best introductory text I know is CATEGORY THEORY by Steven Awodey. Gentle, rigorous and masterly, it's the best book for undergraduates and the only one I'd use for a beginning course in category theory for students that don't have strong backgrounds in algebra. It's pricey, but totally worth it. One other very good-and short-book you should look for and I heartily recommend is T. S. Blyth's CATEGORIES—a terrific short introduction for any student with good mathematics background that wants just the basics in category theory. It's REALLY hard to find now, but if you can get a copy, by all means do so.
That should help you out. Good luck!
A: Sounds like you might want to petition for an exception to the prerequisites.  I don't think Lang's Abstract Algebra is probably your best bet (stick to something decidedly undergraduate -- maybe Gallian?), nor do I think that trying to digest either all of abstract algebra or all of category theory is your best bet.  I'd aim for one major result in abstract algebra which has an analogous statement in a variety of other categories, and then see what carries over to the category-theoretic framework.  One idea would be to understand the classification of finite abelian groups in your abstract algebra work, and try to understand how the result and the proof techniques carry over/generalize.
p.s.  The answer to the title question is definitely yes.  :)
Edit:  Let me add on what I think is almost certainly the best place for you to start on categories, which is Lawvere and Schanuel's "Conceptual Mathematics: A First Introduction to Categories" (double edit: which I see mathphysicist also listed).  In fact, with this book in mind, it's actually the abstract algebra part of your project that now sounds the most daunting -- is that negotiable?  Their discussion of Brouwer's fixed-point theorem would make an excellent topic.
A: Silva, you're studying category theory way too early.  You don't have a background yet that can give you an appreciation for the point of what you're being asked to understand, so probably at best you can follow things line by line (maybe not even that much?) but can't get anything like a bird's eye view of the point of it all. This is like trying to teach abstract linear algebra to someone who hasn't yet had any high school algebra. The motivation is nowhere to be found.  
Ask your advisor what he considers to be some of the important inspiring examples for category theory.  If you don't understand what those examples are, that's a pretty concrete illustration that something is wrong (but then it seems like you already realize it). Then go speak to someone else who can suggest other topics more closely aligned with your background or that start at a more basic level. 
To answer the question, yes category theory gives a lot of insight into the nature of abstract algebra, but only after you've studied enough of the subject on its own for certain basic intuitions (like the meaning and significance of kernel or quotient constructions) to be in your head first. 
