Reference to self-study Abstract Algebra and Category Theory

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.

Given your background, it seems to me that Fraleigh's A First Course in Abstract Algebra would be a great start for a beginner. I've had much success with it in terms of its use in an undergraduate's first course in abstract algebra/modern algebra.

Start with abstract algebra. As Alexander suggested, it wouldn't hurt to work through Fraleigh in conjunction with Dummit and Foote. Then when you get a good "lay of the land", incorporate/begin study in category theory.

See this post: When to learn Category Theory?

See this post: For more suggestions on abstract algebra texts, at variying levels of difficulty.

Here are a couple of free documents that have been created relatively recently that look quite good (disclaimer: I haven't finished either).

Algebra Chapter 0 - Aluffi

Category theory for scientists - Spivak

I would pick up Fraleigh and Dummit and Foote, read them congruently, and forget about category theory until you're done with both of them.

S.Lang, Algebra

S.Mac Lane, Categories for the Working Mathematician