If you want to learn abstract algebra for the first time and you're anything like me, don't just read a book about it. It'll start to sound like a bunch of abstract nonsense. Eventually you'll have the skills needed to "bring it alive" for yourself with a set of concrete examples that you'll learn to refer to again and again. I would recommend starting by watching some lectures about it (you can find some here [1] under the lecture schedule).
He has a very engaging style. I'm about 1/3 of the way through, but had to pause for other commitments. It's mostly possible to track down the homework questions that he sets and solutions for them.
I don't see how a lecture would improve things over a book (assuming you still can't ask questions), though it does of course depend on the book. I'd just pick up Pinter's 'Abstract Algebra'—lots of good concrete examples, one of most readable math texts (in any subject) I've come across.
Edit: also, there's a Dover edition you can get for < $15.
Experienced lecturers might have insights into what kinds of difficulties students typically experience with the material, leading them to organize their lecture in a way that is different from a book. Then again, experienced lecturers might also write books as well where they consider the same things for their book.
It's the difference between something mixing up a 30,000 piece jigsaw puzzle and throwing it on the floor, versus that person laying out a framework, with key pieces where they need to be.
Having a set of concrete examples in mathematics is like unit testing in software. The second you get stuck on a theorem, you fall back on your examples. The second you run out of examples that explain a behavior or new theorem, you write down new examples.
I would diversify my sources. i.e. Sometimes it takes seeing the same thing reworked in another book or in lecture instead of a book to 'get it'. The general rule of thumb is you need at least 3 different solutions or approaches to a problem to really be able to say you know "what's going on."
For a counterpoint, read Shafarevich's "Basic notions of algebra". (Beware: author's meaning of "basic" may differ from own.)
It's a book that explains all those abstract algebra concepts by introducing motivating examples for each one of them. It's the next best thing to finding the natural setting for them yourself.
Agreed. Google images is a surprisingly decent source of visual examples for intuition building/testing, although the signal-noise ratio is a crapshoot.
[1] https://www.math.upenn.edu/~ted/371F14/math371.html