Nodes do not really represent sets; this is again the overspecification that we are trying to rebel against. Nodes represent nodes. Arrows do not represent functions, either.

And a category isn't really a graph, because it's usually too big! So the collection of all nodes is not a set. At best, you could say a category is a Graph (the vertices form a class, not a set), but even that would limit you away from some categories. Best not to think about it!

But for most categories, nodes will be a type of set, and arrows a type of function.

Examples of categories:

Sets
Nodes are sets; we write A -> B if there is a function from A to B.
Groups
Nodes are groups; we write A -> B if here is a homomorphism (of groups) from A to B.
Fields, Rings, ...
Fill in the blanks for yourself
Topological spaces
Nodes are topological spaces; we write A -> B if there is a continuous function from A to B.
pointed categories
If X is a category, Pointed-X is the category with nodes (A,a), where A is a node of X and a a "point" of A; we write (A,a) -> (B,b) if A -> B by an arrow that takes a to b. If we take X=topological spaces, this lets us talk about continuous functions copying some given point to another given point, which is a useful concept in topology, too.

The other important concept in category theory is the functor, which see.