As discussed in the course outline, the textbook for the course will be
I don't have great recommendations for other books. There are other books described as "Foundations of Mathematics" or "Introduction to Proofs" books. I like the Dumas and McCarthy book better than most of these. If you really want to look at another book, you might try Halmos's Naive set theory, or at the notes of Martin Milanič.
We will have a lively online discussion in the UP e-classroom system. Asking and answering questions of your peers will be of great benefit to your development.
Wikipedia has a number of articles on mathematical topics. They are mostly well-written and accurate, and you should find most of the articles relevant to the content of DM I to be fairly readable.