I feel that you are asking two questions. (A) How to get started in research? and (B) What are some good books (for a high school student) in graph theory? Question (A) is in scope for mathoverflow, while question (B) is more appropriate for mathstackexchange. So you should go ask question (B) there.
As for question (A), I think that there are two approaches (and of course you can/should take some combination of both):
"Go to school." That is, go to your mathematics classes, ask questions, learn the material, get a degree, then get another degree, and so on. Talk to the teacher (and eventually the professor), and the other students, to understand what they know, and what they find interesting. Oh, and "school" includes mathstackexchange and mathoverflow and the internet more generally.
"Figure it out yourself." That is, buy a notebook and make a list of questions you find interesting. Write down your own questions. Try to solve the questions of others and yourself. Buy/download a few books that look interesting and spend a year trying to read a few of them (or a few chapters of one book, or a few pages of one book...).
You may find the following questions and their answers useful:
Undergraduate research in Topological Quantum Field Theory
A Learning Roadmap request: From high-school to mid-undergraduate studies