Hubbard's book is by far the most readable for the average good student-I 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 writes, it's crystal clear if compressed within an espilson epsilson of readability.Jost readability. Jost makes up for the density of the text with its clarity.