I think it's going to be hard to find a book that explicates mathematical logic (including propositional logic) without using the notion of a set.
At what level do you want a book on logic? Do you want one that mainly shows how to work IN the propositional (and then predicate) calculus, or do you want one that shows the major theorems ABOUT the propositional (and then predicate) calculus?
For working in the propositional (and then predicate) calculus, I highly recommend 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar. I have found this to be the best combination of rigor and common sense in a practical system for propositional (and then predicate) logic.
My advice would be first to learn a book such as I mentioned, using a common sense notion of 'set'. Then you can apply the predicate calculus to study set theory itself as a formal theory. Then you can use that formal set theory to go back and (this time) rigorously study mathematical logic within formal set theory, so that you you will be studying the formation of (and theorems about) such logic systems as in the Kalish/Montague/Mar as actual formal set theoretic results.
So you might have a plan something along these lines:
'Logic: Techniques Of Formal Reasoning' - Kalish, Montague, Mar
(This will give you all you need to work in the predicate calculus.)
'Elements Of Set Theory' - Enderton
'Axiomatic Set Theory' - Suppes
(This will give you the basic formal set theory you need to study mathematical logic within formal set theory.)
'A Mathematical Introduction To Logic' - Enderton
Hinman, Monk, etc.
(This will present mathematical logic in fairly informal set theory; but with the knowledge you'll have from previous study at this point, you'll see how such informal set theory can be formalized.)
'Set Theory' - Jech.
(As you mention that this book is for later, I agree. I wouldn't use this as a first book in set theory, since it's pretty advanced. Save it for after you've already learned basic formal set theory and mathematical logic.)