For an introduction to proofs you can look at any introductory textbook on (Naive) Set Theory or Elementary Logic. They'll teach you operations defined on sets, the various problems associated with naive set theory, a look at logical quantifiers and negation and methods of proof like contradiction, contraposition and induction. Books like "How to Prove it" by Velleman and "Book of Proof" by Hammack seem to be commonly recommended texts as well.
If you want to get a look at proof-based calculus or linear algebra then "Calculus" by Spivak will be more than enough for the former, and something like "Finite Dimensional Vector Spaces" by Halmos or "Linear Algebra Done Right" by Axler will be great for the latter.
If you're taking analysis in the first year then I'd recommend reading "Understanding Analysis" by Abbott. It's an excellent textbook although I'm using it just as a supplement at the moment.