The topics include rationally connected varieties, rationally simply connected varieties à la de Jong-Starr, unramified cohomology.

There does not seem to exist a single reference for all the topics. Jason Starr has taught a similar course before. Here are the course webpage: fall2018, fall2012.

There are some useful references in Jason's course webpage. There are even problem sets, which students are also encouraged to take a look at, even though some of them are not completely relevant.

Voisin has given a course at Collège de France about rationality problems, the notes are available.

Two useful references for the first part of the course are:

János Kollár, Rational Curves on Algebraic Varieties.

Olivier Debarre, Higher-dimensional Algebraic Geometry.

For the rational simple connectedness part, one could consult several notes of Jason Starr (items 31-33)