There are lots of introductory texts on higher categories, some of which go quite far into things. Check out Rezk's course notes: https://faculty.math.illinois.edu/~rezk/595-fal16/quasicats.pdf ... They're brilliant, and there one of dozens of great bits of writing on this subject.

A friend of mine suggested that instead of a "car" a better analogy is a programming language. Usual category theory and set theory are like Assembly while higher categories are more high level languages (Fortran, C, Python, etc...). These languages were written on top of Assembly, they didn't replace it. Moreover knowing some deep stuff about a programming language (as opposed to just knowing the syntax) gives you more possibilities and power when you're trying to do stuff with it.