What is a good introduction to Teichmuller theory, mapping class groups etc., and relation to moduli space of curves or Riemann surfaces?
The primer on mapping class groups, by Farb and Margalit.
John Hubbard has a recent book on Teichmuller theory which is quite good and geometric.
I find "An Introduction to Teichmuller spaces" by Imayoshi and Taniguchi to be a pretty good reference. Its advantage over Hubbard is that it exists on gigapedia, but I don't know how it compares to the other books in this list.
In addition to the ones already mentioned:
J. Harer's lecture notes on the cohomology of moduli spaces (doesn't have all the proofs, but describes the main ideas related to the cell decomposition of the moduli spaces; Springer LNM something, I believe; unfortunately I'm away for the holidays and can't access Mathscinet to find a precise reference).
K. Strebel, Quadratic differentials (careful exposition of the complex analytic results used to construct the cell decomposition mentioned above; not much about moduli spaces or Teichm\"uller theory though; Springer Erbebnisse).
L. Ahlfors, Lectures on quasi-conformal mappings (construction of Teichmuller spaces).
L. Ahlfors' and L. Bers's papers in Analytic functions, Princeton, 1960.
If you're more analytically minded, I recommend
Gardiner and Lakic, Quasiconformal Teichmuller theory
Nag, The complex analytic theory of Teichmuller spaces
Looking at my bookshelf, there's a few other books that come to mind with varying levels of relevance:
Ahlfors & Sario. Riemann Surfaces.
Jost. Compact Riemann Surfaces.
Maskit. Kleinian Groups.
Tromba. Teichmuller theory in Riemannian geometry.
Farkas and Kra. Riemann Surfaces.
For my own purposes the Hubbard book is what I'd consider a natural starting point.
Hubbard's book is by far the most readable for the average good student -- I don't think it makes sense to begin with anything else right now. When the projected series is finished,it should be the definitive introduction to the subject.
For connections between all these subjects,there's probably no better current source then Jost's Compact Riemann Surfaces. Although the treatment of Teichmuller spaces per se is brief in the book,it contains a wealth of other important topics related to Riemann surfaces. Like everything Jost writes, it's crystal clear if compressed within an epsilson of readability. Jost makes up for the density of the text with its clarity.
Ivanov has a nice review of much of the theory of mapping class groups here. The emphasis is on mapping class groups rather than Teichmuller theory, but the latter is certainly discussed. I find this to be a very useful reference.