Turns out Group Theory's really neat, with lots of pretty diagrams and cool intuitions and beautiful theorems. (Disclaimer, only gone as far as the Sylow Theorems, but still...)
This unexpected revelation came to me through Nathan Carter's 'Visual Group Theory', possibly the most readable maths book I've ever come across.
Despite being readable, I would say 'un-put-downable', it's a proper maths book. You need to do the exercises. Don't go on to Chapter 2 until you've done all the exercises from Chapter 1, etc.
The most common complaint is that it's too slow at the start, too fast at the end. This is not true. The slow bits at the start are building your intuition by playing around with simple cases.
You'll end up being able to flip between algebra and geometry and graph-theoretic ways of looking at groups.
Towards the end, real theorems start appearing, the ones you'd get in a first-year undergraduate groups course, and when you look at them through all the new lenses you've acquired through the early part of the book, they're obvious and beautiful. That's kind of the point of the book.
You're only allowed to complain about how hard chapter 9 is if you've actually done all the exercises in chapters 1-8.
If you find yourself in that sort of position and can make it to Cambridge, get in touch! I can explain everything I've read so far, and will happily exchange in-person supervisions for coffee in pub gardens.
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Other helpful things I've used over the last few months of fascinated exploration are:
The errata page: http://web.bentley.edu/empl/c/ncarter/vgt/errata.html
Actually the fact that there are errors in the book has made it more fun.
Sometimes you find something in the book that seems a bit fishy. After you've thought about it for a while, you can usually make it make sense, but if it still doesn't, it's probably been reported as an erratum by now, so you can go and check. (Also, I managed to get one in myself! The pride! The glory!)
The wikipedia definition of semi-direct product: https://en.wikipedia.org/wiki/Semidirect_product
I could not make head or tail of Nathan's notion of 'rewiring diagram', so I couldn't get more than the vaguest sense of what a semidirect product was.
So I ended up working out how the semidirect product works in an algebraic sense. *That* requires knowing what an automorphism is, which is not a terribly difficult concept, and it turns out that 'rewiring diagram' is an excellent way of visualizing automorphisms.
You can probably shortcut this process by just working out what an automorphism is and how it relates to Nathan's rewiring diagrams.
An online implementation of the Todd Coxeter coset enumeration algorithm:
https://math.berkeley.edu/~kmill/tools/tc.html
It's a very good idea to learn how to do this by hand, and the best way is to read Todd and Coxeter's initial paper.
Todd, J. A.; Coxeter, H. S. M. (1936). "A practical method for enumerating cosets of a finite abstract group". Proceedings of the Edinburgh Mathematical Society. Series II. 5: 26–34. doi:10.1017/S0013091500008221. JFM 62.1094.02. Zbl 0015.10103.
All modern explanations of it are incomprehensible.
However actually doing it by hand gets old pretty quickly, so getting a computer to do it is really useful if you just want to explore.
A page of small-group Cayley diagrams:
http://www.weddslist.com/groups/cayley-31/index.html
