We will start out with some background on surfaces, curves in surfaces and mapping class groups. One unified source for much of this material is Farb and Margalit's
"primer on the mapping class group". You can download it at
www.math.utah.edu/~margalit/primer/ We will not spend a great deal of time on this, however. The main objective of the class will be to go through the proof of
hyperbolicity of the curve complex and the construction and structure of heirarchies of tight geodesics. These have proven to be some of the most useful tools in
studying not only mapping class groups, but also Teichmuller spaces and hyperbolic 3-manifolds. The goal is thus to develop an understanding of the theorems and
techniques of the papers listed above.
There are notes by Saul Schleimer on the curve complex which might be helpful that you can find here.
A nice survey-type discussion of Masur and Minsky's approach is in Minsky's paper: "A geometric approach to the complex of curves on a surface", available on his website . This also has a discussion of the fact that when ξ(S) =1, that C(S) is 3/2--hyperbolic.