Are there any books dealing with concrete categories besides "Joy of Cats"?
I find "Joy of Cats" hard to read and it is not always obvious that the concepts introduced are all that important or how they are important.
Please keep in mind that I'm specifically asking about concrete categories (those equipped with a faithful functor into another category, especially into $\mathsf{Set}$), not just categories.
As an example consider the questions: "How does a left-adjoint to the faithful functor into $\mathsf{Set}$ yield the concept of a subobject generated by a subset?" and "What property in my 'base' category is enough to ensure the existence of enough projectives in my concrete category?". These are (I'd say conceptually) questions about concrete categories, not mere categories.
You can see Kucera's paper here: http://www.sciencedirect.com/science/journal/00224049/1/4 on the topic that every category is a factorisation of a complex one. Hope it helps.