Book about different kinds of logic. I'm searching for a book that talks about different kinds of logic (esoteric and particular ones too) and their uses and differences.
Does such a book exist?

There are plenty non-esoteric, but "less classical" logics.
For example, logics in which you can have infinitely many conjunctions, or infinitely many quantifiers, there are other logics in which you add quantifiers (which may or may not be definable in second-order logic, but usually not definable in first-order logic), these may include things like "For all but finitely many ..." or things like $Qxy\varphi(x,y)$ indicates that $\varphi$ defines a linear order, and so on and so forth.
Many of these are covered in the book Model Theoretic Logics (eds. Barwise, Feferman) which is available freely.

Another good point to start with would be Stanford Encyclopedia of Philosophy: classical logic. Yes, this is about classical logic, but in the related entries section you can find many non-classical which are more or less esoteric, including references and possible internet resources.
