This is especially true with graduate level algebra, most of its theorems are so convoluted (not as beautiful as those in the computational math I used to have) that left me wondering: "What is it all about?" and "Did anybody ever use this theorem before?"
You might find this interesting:
http://pauli.uni-muenster.de/~munsteg/arnold.html
Maybe slightly over-stated, but hey, that's part of the fun. I always sort of "knew in my heart" that the way math is taught is often wrong or at least missing something. Many very high level mathematicians at least partially agree with me. Advanced math textbooks are often a good reference for the final form of a theory and as a source of exercises to practice on, but they tend to be pretty weak at conveying an idea of what the subject is all about and why anyone would care, and it can also be difficult to extract the intuition from them.
Unfortunately, most of the books on abstract algebra that I suspect are good for that purpose are books that I have not actually read. Most of my own ideas of what abstract algebra is about come from using it in other subjects and thinking about the significance of the definitions and theorems for myself.
I'll give you the list of algebra books that I should have, but have not read, here:
Symmetry by Hermann Weyl
Visual Group Theory by Nathan Carter
Elements of Algebra: Geometry, Numbers, and Equations by John Stillwell
A Book of Abstract Algebra by Charles Pinter
Abel's Theorem in Problems and Solutions: Based on the lectures of Professor V. I. Arnold by V.B. Alekseev
I'm least confident about the Pinter book because I'm not familiar with that author, but the rest seem very promising, based on who the authors are, and things I've heard or read and so on.
Here are a few more that I have read parts of:
Galois Theory by Ian Stewart
Abstract Algebra: The Basic Graduate Year
http://www.math.uiuc.edu/~r-ash/Algebra.html
There's also a number theory book by Stillwell that's a companion to his algebra one that I mentioned, Elements of Number Theory.
Another book that's more of a special topic, but involves amazing geometrical intuition that would help with group theory in general is this:
Mirrors and Reflections: The Geometry of Finite Reflection Groups. It assumes you already know the basics of group theory, but it would be a great place to apply it in a more geometric context. Personally, my interest in the subject comes from the fact that it clears a lot of the fog surrounding the classification of Lie Groups/Algebras that I studied in grad school, and found lacking any intuitive glue to hold it together and make it memorable or explain where the results come from. I gave up on trying to do math research, but it's my hobby to try to put right a lot of the things that were thrown at me in grad school that were lacking in motivation and intuition.
I originally learned graduate algebra from Dummit and Foote, which has a lot of nice aspects to it, but in some ways falls into the trap that I mentioned at the beginning of not having the best motivation.
You can also consider looking into the history of math. Morris Kline's book is one example. He talks about the origins of some of the concepts in algebra.
If you are interested in applications, you should consider looking into coding theory, cryptography, and possibly physics or crystallography.
Another thing to keep in mind is, as I mentioned, sometimes learning other branches of math makes things about abstract algebra click more. If you study differential forms from a good source, then stuff like exterior algebras might make more sense, although if someone explained it well enough, you wouldn't really need to know it. Lots of stuff can be that way. Makes more sense in a different context. From topology, there's the notion of a fundamental group, which is another way to think about groups.
So, you can see that there's a lot of stuff to read if you want to get to the bottom of "what the subject is all about", which might be one reason why some mathematicians are tempted to skip most of the fun stuff and stick to the more boring, formal aspects. The good news is if you read everything I listed, there would probably be a lot of redundant material.
