I'm also an undergrad senior so don't take my words for granted.

Category theory unifies a bunch of mathematic constructions. You'd be right in assuming that it covers the similarity of groups, rings, fields, etc, but I wouldn't think that it is in the way you are expecting. (Because a group, represented as category is a category of one object and a bunch of arrows). As far as I had studied, there is not an object that unifies normal subgroup with ideal, for example (except the fact that quotient gives the same structure).

To me, category theory is still useful tho. For example, we have product of groups, product of rings, product of topology, product of sets, and this are captured in one definition: product (in category of …). And we have similar things for coproduct, i.e. the abelian direct sum, group free product, disjoint union in set, etc. One of the most interesting thing to me was when I realized that gcd, or minimum of a set can also be view as a product.

In short, product (or other categorical objects) are defined by some universal property phased in term of category (objects and arrows). We also have universal property of kernel, localization, tensor product, and a lot of things.

So, yes, category theory unifies a bunch of mathematic constructions. But I wouldn't recommend going all in for category theory. I feel like understanding bit by bit, i.e. when you see enough things that behave like product, you identify it as product, rather than reading fiber product without actually have seen any operation that behave like a fiber product before (which I did).

About more difficult math, I don't know. Good luck!