I have learnt that first First-Order Logic (FOL) provides a formal language for set theory.
That given, one would expect that, when defining the symbols and rules of FOL, textbooks authors would do so without sets. However, textbooks I met use sets to explain FOL. This being the case, it make less sense to use FOL for set theory, because I already need the primitive notion of set to speak about FOL.
Initially, I thought that using the set-free approach was technically unfeasible. That's until I found a free ebook, doing so: An Introduction to Set Theory by Professor William A. R. Weiss.
Is there a particular reason for following the other approach (FOL based on set, then introduction to axiomatic set theory)?
I would also read more in the Weiss style. Can you give me some references about books not using sets when explaining the formal language? I tend to prefer textbooks as references (rather than seminal papers/books).
There is a pretty good introduction to First Order Logic and it is an open source book provided by P.D. Magunus. As you say, Truth Funcional Logic and First Order Logic are defined without using Set Theory. I leave you here the link: https://www.fecundity.com/logic/download.html