The (unpublished) book "Introduction to Higher Math"[1], along with the professor who taught the course, was probably what got me going on the road to a PhD in math. It's very accessible (no prerequisites to speak of), teaches the reader how to write mathematical proofs, and includes some really interesting material, e.g. cardinality. It's also freely available now.
[1] https://www.whitman.edu/mathematics/higher_math_online/