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.