You are observing the evaluation map ev_p: C(M, R) -> R, ev_p(f) = f(p), is a ring homomorphism. In that spirit, you might find this Terry Tao article on the Yondea lemma interesting: https://terrytao.wordpress.com/2023/08/25/yonedas-lemma-as-a...

My heart goes out to all the math students who were frantically texing their problem sets last minute

Richard Borcherds, a Fields medalist, has been putting out great math content on YouTube for the past year and a half (https://youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ). He even has lecture courses on a variety of undergraduate and graduate classes.

his sense of humor is impeccable, too

I like the way the Functional Programming HOWTO in the Python docs (https://docs.python.org/3/howto/functional.html#small-functi...) puts it:

  Fredrik Lundh once suggested the following set of rules for refactoring uses of lambda:

  1. Write a lambda function.
  2. Write a comment explaining what the heck that lambda does.
  3. Study the comment for a while, and think of a name that captures the essence of the comment.
  4. Convert the lambda to a def statement, using that name.
  5. Remove the comment.

Alex Wright recently wrote a nice mathematical exposition on some of her work relating to Riemann surfaces, targeted at "a broad audience of non-experts."

https://arxiv.org/abs/1905.01753

Another great explanatory paper written for a broad audience is http://www-personal.umich.edu/~alexmw/BilliardsToModuli.pdf

A nice foundational textbook to start with is Michael Spivak's text "Calculus." This text is fairly conversational, it motivates its concepts well with many examples, and it will help you build a strong foundation in writing proofs and reasoning mathematically.

As others have said, it might be helpful to have a friend or two read with you so you can answer each other's questions. Also, make sure you do a ton of exercises!