|\N| = |\Z| = |\Q^2457000| != |\R| isn't wondrous to you? Nor are space-filling curves? Or unmeasurable sets and paradoxes like Banach-Tarski? You don't see beauty in the proofs of Cantor's theorem (|S| < |P(S)| for all S) or the Bernstein-Schroeder theorem (injections X -> Y and Y -> X imply |X| = |Y|)? What about the plethora of mathematical claims that are equivalent to the axiom of choice -- some of them very sought-after properties, some of them unnerving and seemingly broken? Does the status of the continuum hypothesis (and the mathematical legitimacy of, to say nothing of the resolution to, a bunch of oddball "infinitary combinatorics" problems) interest you?
What about where set theory interplays with model theory? Would it surprise you to hear that the dizzying towers of infinities that can be spoken of in ZF can be modeled by a merely countable universe? What about a computer program (albeit an impractical one) that decides whether sentences about the natural numbers with successor are true or false, whose existence is discovered by proving that large uncountable models are isomorphic?
To be clear, it's okay to be bored by this stuff. I tend to get bored pretty quickly when people start talking about systems of DEs or eigenvectors, or really much of anything that looks too much like calculus, and I'm not about to apologize for my apathy for those things. But most of the above stuff tends catch the attention of even the humanities types, who are easily deterred by most mathematical topics.
Out of curiosity, I might ask whether you're similarly bored by computability theory or complexity theory, which leverage many of the same tools as those used in set theory (mappings as preorders, diagonal arguments, etc), but add computable enumerability/uniformity/syntacticity requirements to the mix.