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In my undergraduate days I was at best a fellow traveler alongside bona fide combinatorists, Ramsey theorists, and graph theorists, but I was never a true apprentice of any one of them number fields exerted too strong a pull. However, frequent exposure did have its effects and I came to understand that there are a host of beautiful results in graph theory, often remarkably easy to state and even illustrate, but ever so difficult to prove.
Graph theory shares this property also with combinatorics as well as with elementary number theory and it has a lot of overlap with these fields, of course — so much so that it is sometimes difficult to say where one discipline ends and the other begins. Graph Theory aims to answer this question at least at the level of a beginner in the field: there is a well-defined core of material every self-respecting graph theorist should know, and a useful graduate-level textbook goes a bit further by introducing what should or might loom on the horizon as research in the field progresses.
So there it is: if you want to be a graph theorist, learn all this stuff well — and, to be sure, this is a superb place to learn it. Even the pictures and drawings are nice.
This is a hell of a good book! Unlike, say, algebraic geometry, graph theory is one of those marvelous disciplines which are very beautiful and serious but at the same time get off the ground very quickly.
An able and energetic fledgling would be safely pointed in this direction, and should be given this book as standard issue. The Basics.