The best way to think about “what is Scalaz?” is as a standard library for functional programming. This goes all the way back to its creation: the Scalaz project started because there are not enough facilities in Scala's standard library for convenient, everyday functional programming, without cheating.
How should this affect your approach to the library? Like a standard library, you learn the bits and pieces you need, not the whole thing. There is no must-read book, no must-watch tutorial video, no must-attend course. Scalaz can be used successfully from day 1 as a new Scala programmer; as you do not learn every part of the standard library before starting to use a language, so it goes for Scalaz as well. All that is required of you is the desire to solve programming problems in type-safe, functional ways, and the curiosity to learn about what components that others have discovered and how they might be useful. After all, most pieces of Scalaz were added to it because somebody was solving a problem, and found a solution they thought others might consider useful and well-thought-out.