Are there good books or lecture notes just about 2-categories?(not about higher categories nor $\infty$-categories)
(I'm studying fibered categories for the descent theory of quasi-coherent sheaves. I want to understand fibered categories from 2-categorical view point.)
This by Vistoli might be what you're looking for, especially chapter 3. For more discussion on 2-categories, see here. There are some more good resources in the references section of https://ncatlab.org/nlab/show/2-category.