What are good books to learn graph theory?
Try Bondy and Murty, Graph Theory. The previous version, Graph Theory with Applications, is available online.
Diestel's Graph Theory (which has a "free preview" online) is presented as a graduate textbook, but it does not really have any prerequisites.
It goes quite deep in some parts, and includes material (such as the chapter on the graph minor theorem) that you won't find in other textbooks. Some proofs have a sort of "how would someone ever think of that?" feel, but this may have been remedied in the fourth edition (I have the third).