Lecture 1: Spectra and Linearization

Lecture 2: Dual Steenrod algebra. See Eric Peterson's book (section 4.1) and Jacob's lecture 13 from the Sullivan course.

Some people might find this little tutorial on some things in infinity-category theory helpful: tutorial.

Lecture 3/4: See Symmetric monoidal stuff and also this paper. See Nikolaus-Scholze for more on the Tate diagonal.