I just finished chapter 3 of the book and I'm not sure whether I'm doing it right.
The chapter has 10 pages. After the chapter there are 92 exercises.
Do people do all of these or do they pick some? How would you know which to pick if you don't have a teacher telling you which ones you should do?
On
When using Gallian to self study abstract algebra, I found it helpful do do 10-15 exercises before moving on. I did any exercises that asked to prove part of a definition in the chapter, and I also did a 3 or 4 computations and 3 or 4 proofs that required ideas from the chapter. After that, I moved on to the next chapter. If I felt like I had difficultly remembering a topic from a previous chapter, I went back to the chapter and worked through some more of the exercises.
You could certainly work through all of the exercises, but after a while, many of the exercises use the same types of proof techniques, and doing more exercises won't improve your understanding of the definitions that much.
Gallian (2nd edition) was the book that first got me interested in abstract algebra (when I was an EE grad) and pure mathematics in general, and I went on to get a PhD. I know he's well beyond the 2nd edition now, but at the time I did every exercise in the entire book and found it very worthwhile. (I'll admit though that 92 exercises is rather a lot.)
In general, I feel like when you're learning a subject it's useful to do every exercise. The harder ones will teach you something new, the easier ones are good practice in the basics and learning to communicate your ideas precisely and succinctly.
Solving the problem to your own satisfaction is the essence of mathematics, but communicating it to others is arguably what makes mathematics so powerful - that we can convince others of the veracity of the statements we make.
With so many exercises, though, I would say maybe it's enough to read each one, identify the ones that you can do in your head and just do that, but anything you can't do in your head is probably worth spending time on and writing out. Gallian does gives lots of exercises involving very specific examples, which are great for "getting your hands dirty". As you progress towards graduate level texts these are less common and most exercises/problems ask you to prove things in some generality.