TITLE: 2-category theory
QUESTION [13 upvotes]: I know that we can do a lot of 2-category theory, seeing 2-categories as Cat-enriched categories. Yet, I know that there are some limitations of this approach.
I also know that there are many articles which could help to understand 2-category theory... (I am only familiar with a few of the Lack's, Street's and Kelly's articles so far, but I know there are many more important articles).
But always when I'm trying to deal seriously with 2-categories, I end finding serious difficulties.
So I am sure I have to improve my 2-category theory knowledge in general. And just recently I have become aware of the Gray's Formal Category Theory. So my question is about basics of 2-category theory: which set of articles or books could be considered as a solid base to start thinking seriously about 2-category theory?
I am just trying to avoid a common situation for me: being stuck in a well known (and basic) subject of 2-category theory, ignoring the existence of (classical) literature about this subject.
Thank you in advance

REPLY [4 votes]: It sounds like the new book from Johnson & Yau "2-Dimensional Categories" might be what you're looking for: https://arxiv.org/abs/2002.06055
While this is not a classical, or perhaps that well-known reference, it does seem to be a comprehensive and detailed take on the subject