I am a beginner in studying centroid, circumcenter, incenter, and locus of a point. Is there any book that could make me understand their fundamentals well and be able to solve most of their proofs? and then to be able to solve IMO problems?
And what is the best way to be good at mathematics? What books should I study in each field: algebra, geometry, number theory, algebraic geometry, and combinatorics?
@AmirHossein already gave you a link above to a list of books of several fields. So, I will refrain from pointing you to other books. I will only try to answer your final question "And what is the best way to be good at mathematics?":
In order to become good at mathematics you have to do mathematics. This means, you have to try to solve problems, you have to fail solving them, give it yet another try until you ultimately come to a solution.
In some respects mathematics is like swimming: you won't learn it by watching other people doing it. You have to jump into the deep oceans of mathematics and try it yourself. Even though it might be frustrating much of the time, it is well worth the effort. You will end up diving far under the surface and you will be able to see a beautiful world which is hidden to most people. I can only encourage you to jump into the water and start doing mathematics.