In the spirit of being ambitious, I have enrolled in a topics course in Differential Geometry with none other than Mikhael Gromov. I have a strong(ish) background with manifolds and a mediocre background with geometry on manifolds. The course description reads as follows:
"I start with definition and basic properties of the scalar curvature and present a list of relevant examples. Then I will explain basic constructions: gluing and surgery. With this I will prove some classification results for simply connected manifolds with positive scalar curvature. Then I will explain the Schoen-Yau method combined with symmetrisation and prove basic inequalities for manifolds with boundaries. After that I will expose the basics on spinors and on the index theory for the Dirac operators followed by application to scalar curvature."
I am looking to read up on scalar curvature, and am wondering where I should start. I am looking for references that will be not entirely incomprehensible given a pretty thorough knowledge of - say - the entirety of Lee's Smooth Manifolds.
You might find the relevant sections of Lawson and Michelsohn's Spin Geometry a useful resource during the course (maybe not so much as a precursor). In particular, it includes a discussion of the Atiyah-Singer index theorem and its applications to questions about positive scalar curvature. It also includes a very important and well-known theorem by Gromov and Lawson which states that surgery in codimension at least three preserves the existence of positive scalar metrics.
As for learning about scalar curvature, any book on Riemannian geometry will do. For example, take a look at the books by do Carmo, Lee, Petersen, etc.