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:
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.
The exercises in the early chapters of the book by Arbarello Cornalba Griffiths and Harris are very interesting. The book itself is a second course but the early chapters and execises are a recap with interesting side trips.
You can also look at Clemens's book A scrapbook of complex curves, somehing like that.
These answers seem to have almost nothing on Riemann surfaces. I guess I am just too old-fashioned. In a first course on Riemann surfaces, I would like the student to get an understanding of the Riemann surface for log z, and for arcsin z, for example.