I was about to begin trying to work through Elias Zakon's Basic Concepts of Mathematics, available free online at here. My motivations are similar to yours, and the idea of a Remedial Proofs Book Club is attractive to me. I chose this particular text only because it was free online and seemed to be written specifically for people at my level (got a 5 on the Calculus AB AP exam after high school, thought I was hot shit, took a rigorous Calculus course my first of year of college, couldn't do a single problem, cried real tears, traumatized). If you had a strong preference for different source material, I would switch to it if we could work through it together.