The students are aware of mathematical logic and proof but have not come across any of the notions of a set. What is the most natural and motivating way to introduce set theory?
The student background in proofs and logic is not necessary for initial explanation/motivation of the concepts. And actually it worries me that you mention that when wanting to know how to do an initial introduction! I remember learning the basics of sets (and union, intersection, etc.) as an elementary schoolkid in the New Math. You do not need a huge background to start looking at the topic. This is very different from calculus where you really do need some background in algebra, functions, analytic geometry, etc.
The easiest way to first explain the concept is as a collection of things (e.g. stamps, books, etc.) see pages 5-6 of Tenenbaum Ordinary Differential Equations.
Of course you should move to generalize to infinite sets and more difficult concepts. And the students have background to handle that. But to START, just do exercises with simple sets of integers. E.g. (1, 3, 5, 7, 9) versus (2, 4, 6, 8, 10) versus (3, 7, 11, 214) versus (3, 5, 7, 1, 9). I recommend using integer collections as the starting point as it is easy to look at them and discuss union/intersection. Of course, you can move to more fancy complicated things. But start simple.
You can illustrate with Venn diagrams also.
Of course, given the level of the students you can move to some formal proofs as well. AFTER they have been introduced to the topic. but I would not START any topic with a definition, theorem, proof, example style of teaching.
P.s. I also suggest looking at some books (hard and easy) for their approaches. This should not just be something to be solved de novo by a question here. [Or with an approach of constructing a course from scratch. Anyone who DOES construct a course from scratch should have some deep perspective before doing so...and I don't mean deep math perspective. I mean in addition to that having seen different pedagogical approaches and having some formed ideas on what works/doesn't.]