I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold theory, but may not have had much expose to algebraic geometry. I will loosely follow the book Introduction to Algebraic Curves by Griffiths. In particular, I hope to spend a minimum amount of time developing basic machinery (e.g. sheaf theory) and to start doing concrete geometry (e.g. canonical models of curves of genus up to 4) as soon as possible.

My question is: what are some good concrete, accessible geometric topics in Riemann Surface/Curve theory that aren't in the standard textbooks?

Let's say that the standard textbooks are the book I mentioned and those discussed: here.