I am trying to teach myself the basics of Riemannian geometry (curvature, Gauss-Bonnet, etc...). I am having a hard time getting started because I can't build good examples by myself when it comes down to understanding the definitions and the theorems.
For instance, I would like to understand the Gauss-Bonnet formula for compact surfaces without boundary $$\int_M\mathcal{K}\,\mathrm{d}A=2\pi\chi(M)$$ by performing explicit computations of the LHS for, say, the real projective plane.
Could you help me find references (lecture notes, blog posts, textbooks, ...) where this kind of computations for concrete examples is carried out with great detail at the undergraduate level?
Edit: The class notes Differential Geometry: A First Course in Curves and Surfaces (available here) by Ted Shifrin are a perfect example of what I'm looking for.