I would like to know more about relationships between the mapping class group of an orientable surface with negative Euler's characteristic and moduli spaces. In particular, I would like to have a rigorous explanation of the following three facts:
having a family of surfaces given by a fibration $F\to E\to B$ (e.g. take B as $S^1$ and F as a compact surface of genus $g>1$), how to construct a map from $\pi_1(B)$ to $MCG(F)$.
how point 1 should give a relation between the $MCG$ and the topological type of $E$
how the cohomology of the $MCG$ is related with the cohomology of the moduli space of curves
In other words, I know just some chatting about that, but I would like to learn more deeply how to formulate and prove those facts.
Thanks a lot in advance, bye!
Just to leave answers as answers:
There is a new 500$+$ page book by Benson Farb and Dan Margalit called A Primer on Mapping Class Groups. I am not an expert in this area, but I'm not the polar opposite of one either, and the book looks to me to be very strong and likely to become a standard reference. So anyone who is at all interested in this subject would do well to download a copy of the book from the second author's webpage while free copies are still available (I have no reason to believe that a copy of this version of the book will go offline once the final version is published, but it doesn't hurt to be safe).
Added: the OP has made clear that his specific questions are not addressed in this book, so this is a very partial answer (more of an answer to the title than the question itself). I hope others will contribute more pertinent information.