Why do professors go through proof after proof with no rhyme or reason?
One theory is that this is an "easy" way to give a lecture (to be negative about it, a "lazy" way.) This may be true in some cases. But on the other hand, much of the instructor's education might have been this way, and maybe they even think the experience is valuable. So, they might actually be giving the students the best path they know of. Some students might even feel like that is the way they are most comfortable learning. So to be fair, such instruction may be given in good faith, and may have good points.
The fact is that really good exposition requires a really skillful teacher, and it's not easy to do. Incidentally, I found Artin a very good expositor, but I did observe that by doing this, some less dedicated readers might get bored or distracted during his exposition.
One of my books learning abstract algebra was Martin Isaacs' Algebra. At the time I did not like it very much, but looking back on it now I think I do like its exposition. This just goes to show that reasonable exposition is not always easy to evaluate.
My question is How do I prevent this from happening to me in future?
Oh, well that's easy! Go skim through a lot of alternative books on the same topic and soak up whatever you can! Don't pretend like it's your teacher's responsibility to put text on your plate. You already applied this when you picked up Artin's book and learned something from it.
why are textbooks like Gallian's popular in math instruction?
The "like" part here makes this a loaded question, but I could just say that this book is probably considered basic, safe and affordable. It probably also depends upon the teacher's experience with texts too.
How are you supposed to read them?
This varies a lot from person to person. Personally I discovered that I learn best by having three or four texts on the same topic that I can use to cross-reference topics. Usually at least one of the authors is going to say something that makes things click.
And most of all, this sets me up with a big supply of problems. Doing problems does a lot more than plain reading, for me. Of course you have to spend some time reading or you won't know what tools you have at hand, and you won't see the themes in the proofs.
Is math supposed be learnt like you are learning a game where you are given all the rules ( definitions) and you have to play the game according to those rules( theorems, exercises) ?
I guess ideally "no", but for some people, that's how mathematics first begins! Those who persist eventually find their own appreciation for the subject matter, and develop their ability with it. This "game" analogy certainly doesn't paint a pretty picture of pedagogy, but it's very rare to find teachers with enough ability to get the beauty of mathematics across from the very beginning.
Luckily, it sounds like you at least know mathematics is more than a string of memorized definitions and theorems and proofs, so you, my friend, are already well ahead of many other students. The rest are in the even sadder situation of thinking "Yes, that's all mathematics is. Isn't it awful?!"
Best Answer
I'm not an expert in the history of ring theory but this is, I think, pretty close to a correct answer:
You are right that the notion of "prime integer" predates the more general notions of "prime element" and "irreducible element" in an arbitrary ring. In fact, prime numbers go back to ancient Greece! But there is a missing link in the evolution of that original notion into the (two distinct) modern notions: namely, the notion of a prime ideal.
Ideals were regarded as a kind of "generalized number"; in fact, the original terminology was "ideal number", only later shortened to "ideal". One ideal $I$ was said to divide another ideal $J$ if and only if $J \subset I$. A prime ideal is then defined, in precise analogy with the "classical" definition of prime numbers (i.e. as indecomposables) to be an ideal that is not divisible by any ideals other than itself and the entire ring.
Once "prime ideal" was defined, the next development was to say that an element was prime if it generated a prime ideal. It is a fairly straightforward exercise to show that this translates directly to the modern definition of prime element. It is also fairly easy to show that (as long as there are no zero-divisors in the ring) every prime element is indecomposable in the classic sense. So everything fits together quite nicely.
It is only at this point that somebody starts looking at rings like $\mathbb{Z}[\sqrt{-5}]$, which are not unique factorization domains, and realizes that those rings can contain elements that are indecomposable in the classic sense, but do not generate prime ideals. Whoah! So we need a name for those types of elements. "Prime" is already taken, so they get called "irreducible".
So there you have it. The elements that we now call "irreducible elements", despite the fact that they have the property that we usually associate with "prime numbers", were not called "prime elements" because that word was already in use for elements that generate "prime ideals", which are defined in direct analogy with how we "usually" define prime numbers.