You might be interested in the book Graphs, Groups and Trees by John Meier. It is a very readable introduction to "geometric group theory", and is pitched at "advanced undergraduate" level (in the question linked in the comments, some of the books are graduate level and above). Geometric group theory can be interpreted as "the study of groups using their actions", and the simplest actions are actions on graphs. Hence, this book studies groups by using their actions on graphs.
Topics covered in the book include group actions, Cayley graphs (every group acts on a graph, and the Cayley graph is such a graph), actions on trees and basic Bass-Serre theory, the word problem for groups, regular lagnauges and normal form, and the coarse geometry of groups. There are lots of examples to get your teeth into (every other chapter takes a specific group and analyses it using the previous chapter).
I should say that Meier's book assumes a working knowledge of group theory, but I would be surprised if there existed a book on this subject which did not!
@ABajaj The book you were reading, by Grossman and Magnus, was from the "new mathematical library". This "library" was a collection of books pitched at your level (US high school) which accompanied a new method of teaching maths (called new math) in the US. The method was generally considered a failure, and therefore I would doubt if there was a set follow-on book. Meier's book perhaps requires a small jump from where you are just now, but this jump can most likely be helped by also buying an introductory group theory text to use as a reference. If you know what a normal subgroup is, then you are probably good to go! (The word "Sylow" does not enter Meier's book, although enters every single "standard" group theory text, so your jump will not be too big!)
I should point out that the name of Magnus is a famous one in geometric group theory. So Meier's book, and geometric group theory in general, is a natural place to go next.
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The book categories for the working mathematician, by Saunder's MacLane, comes well-recommended. Although I have never quite got around to reading it, it has such a promising title! Note that Saunder's MacLane was one of the original category theorists.
For an explicit example of graph theory and category theory working together to prove results on groups, there is a well-known link between the category of graphs and subgroups of free groups. The classical reference is Topology of Finite Graphs by Stallings. For example, if the non-diagonal component of the fibre product of a graph with itself is simply connected then the corresponding subgroup is malnormal. Dani Wise lifted this idea to the more general category of "cubical complexes", which allowed him to prove some famous open problems in group theory and G&T (for example, he proved the virtually Haken conjecture, and that every one-relator group with torsion is residually finite). I found Wise's paper The residual finiteness of positive one-relator groups to be especially helpful.
You say you want to use category theory and graphs to look at endomorphisms of groups. Well, one of the questions I was attacking in my PhD thesis was the following.
I fixed a class $\mathcal{C}$, and I managed to prove that the above result held for this class so long as I could prove that a certain group (in reality, a class of very similar groups) had a malnormal subgroup (with certain additional properties). So, I then took the fibre product of a "subgroup" in the ambient free group and (with a bit of effort) proved that its malnormality fell down to my group.
This doesn't quite answer your question, but I thought it relevant enough to mention. If you want, I can send you a copy of my thesis.