As Euclid has shown in his Elements that he can combine and construct entire understanding about a subject by simply using a few axioms. I am wondering, if I want to study a new subject, let's say group theory, is there any idea that is like this kind of "axiomatic approach" when studying a new subject?
In other words, if I really want to learn a new subject from the most fundamental causes, take group theory as a example, can I treat those definitions in group theory as the "axioms" that I can use to combine and construct in my study process? If not, what's the disadvantage of doing so?
In real life, the axioms for group theory came after people already understood many examples of groups, and their behavior. Likewise, axioms for geometry only came after people had extensive physical experience with geometric aspects of reality. We don't make random sets of axioms, but, rather, (at most) sets of axioms that are meant to capture various prior experiences.
To the secondary question about how a person without prior experience should approach a subject, axiomatized or not: examples. Phenomena. In many standard "abstract algebra" sources (in which introductory "group theory" lies), one should mostly ignore the lemmas and propositions and theorems, and look at examples, some of which should be in the text, but/and many of which will be (somewhat deceitfully) relegated to exercises. The latter relegation is considerably unfortunate, because it creates the impression that knowing definitions and theorems should enable us to understand all the examples. In my experience, it's more the opposite. :)