What should be taught in a 1st course on Riemann Surfaces? 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. 
 A: As Gerald said, really understanding some specific surfaces is useful.  And not just the compact ones!  Remember that Riemann's ideas were based on analytic continuation, not algebra.
And if you really mean Riemann surfaces, then Divisors and monodromy (compute some actual monodromy matrices!).
And if you want to cover material which is not in the standard textbooks, cover in depth the relation between Riemann surfaces and differential equations.  
A: 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.
A: One thing that I rather like (though I'm biased, and mentioning old work by my advisor...) is the theory of Prym varieties, which can be mentioned immediately after discussing Jacobians.  In particular the $n$-gonal construction (see Donagi "Fibers of the Prym Map") has a lot of nice geometric consequences, including a proof that the intermediate Jacobian of a cubic threefold isn't the Jacobian of a curve (though that's not really a Riemann surface thing), but generally, Prym varieties are rather nice (as well as Weil pairing and theta characteristics) but aren't mentioned in most courses...I think they've got short appendices towards the end in Arbarello-Cornalba-Griffiths-Harris, and they leave out most of the details.
A: Puiseux series and the Newton-Puiseux theorem are beautiful and very useful to understand ramification and related issues. They do appear in one of the "standard textbooks" of the list (Farkas-Kra) but it seems they are usually overlooked.
A: Although it is sort of indirectly related, it might be nice to talk about some introductory abelian variety things (as in the first few pages of Mumford's Abelian Varieties). The motivation would come from proving the equivalence of the definition of genus as the dimension of the Jacobian variety of the curve. When I took a "curves" class, I would have liked to see this rather than thinking the course was "self-contained".
Do not be afraid to show glimpses of huge areas of math that were motivated by the study of curves, even if you don't have time to do more than just mention it. I would have been far more excited and motivated to learn some of these things if I had seen it as motivated by curves, rather than the other way around (studying abelian varieties as interesting in their own right and only later learning a motivation).
A: 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:

*

*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.


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


*Weierstrass points, with applications to 2) above.
A: 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.
