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