I'm looking for a modern book on logic-set theory-foundation written as the Bourbaki's set theory. I'm particularly interested in a formal exposition of ZFC axiom with logic-set Grothendieck universe.
Bourbaki's book was idiosyncratic, and the terminology it used was never common. Contemporary books do not use terminology such as "assemblies"; they talk instead about "formulas" and "sentences".
The two most common contemporary books on set theory are Set Theory by Kunen and Set Theory (3rd edition) by Jech. Kunen's book is written in the style of a textbook, while Jech's is written in the style of a desk reference, but each book can serve both purposes. Both of these books start at the beginning and present set theory in a fully rigorous manner.
The existence of Grothendieck universes is equivalent to the existence of strongly inaccessible cardinals. Inaccessible cardinals are very well understood in set theory, so set theorists don't spend much time also talking about Grothendieck universes.