ag.algebraic geometry - What should be taught in a 1st course on Riemann Surfaces?

Good question. I bet you'll get many interesting answers.

About two years ago I taught an "arithmetically inclined" version of the standard course on algebraic curves. I had intended to talk about degenerating families of curves, arithmetic surfaces, semistable reduction and such things, but I ended up spending more time on (and enjoying) some very classical things about the geometry of curves. My lecture notes for that part of the course are available here:

http://www.math.uga.edu/~pete/8320notes6.pdf

Some things that I found fun:

1) Construction of curves with large gonality. For instance, after having given several examples of various curves, it occurred to me that I hadn't shown them a non-hyperelliptic curve in every genus g >= 3, so then I talked about trigonal curves, and then...Anyway, there is a very nice theorem here due to Accola and Namba: suppose a curve $C$ admits maps $x,y$ to $mathbb{P}^1$ of degrees $d_1$ and $d_2$. If these maps are independent in the sense that $x$ and $y$ generate the function field of the curve (note that this must occur for easy algebraic reasons when $d_1$ and $d_2$ are coprime), then the genus of $C$ is at most $(d_1-1)(d_2-1)$.

I sketched the proof in an exercise, which was indeed solved in a problem session by one of the students.

2) Material on automorphism groups of curves: the Hurwitz bound, automorphisms of hyperelliptic curves, construction of curves with interesting automorphism group.

3) Weierstrass points, with applications to 2) above.