What should a student (with algebraic-geometry minded) study in differential geometry?

One of my friend who is an undergraduate student, has known something about algebraic geometry (equivalent to chapter 1 and a little bit chapter 2 in GTM 52 by Hartshorne). He is now has to study a differential geometry course. Since there are a lot of Differential geometry text books with different intention, he asked me what he should learn in differential geometry so that it helps him to improve his knowledge in algebraic geometry

To start with the basics, the strangest thing in differential geometry for somebody coming from an algebraic-geometry background is the idea of partition of unity. So this would have to be internalized as an introduction to a DG point of view as opposed to AG. Next, I would suggest performing the following thought experiment: take $\mathbb{CP}^2$ blown up an $k$ points. Now reverse the orientation of this beautiful algebraic variety. Does the resulting manifold make any sense?