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.
Best Answer
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.