Freyd's Abelian Categories is the only textbook I know where the primacy of arrows over objects is taken seriously already in the axioms: there is no talk of objects at all. Only later one sees, that an object is what arrows with the same domain (another derived-only concept) have in common. This helps thinking abstractly from the very beginning.
Question #1: Why isn't Freyd's axiomatization standard?
I guess it's mainly a matter of convenience, but this seems to be rarely mentioned explicitly.
The same goes for graph theory: the usual and beginner's way of thinking is that arcs (seen as pairs of vertices) are 'ontologically' secondary to vertices, just as arrows (seen as functions between sets) would be secondary to objects (seen as sets). But in the most general setting for graph theory - multidigraphs or quivers - it becomes obvious that it is the other way around: arcs can be the only primary objects and vertices can be thought of equivalence classes of arcs according to the two equivalence relations "has the same source (resp. target) as".
Question #2: Is there an introductory textbook to graph theory that emphasizes that general graph theory is "nothing but a theory of two arbitrary equivalence relations".
[Addendum:] The structure of a graph comes in by identifying some equivalence classes of the first equivalence relation with some of the second.