One of the best introductions to the subject is certainly Thurston's [Three-dimensional Topology and Geometry, Vol.1][1] (not to be confused with his much harder lecture notes *Three-dimensional Topology and Geometry*). It has almost no prerequisits, but leads you right to the statement of the geometrization conjecture of Thurston and some surrounding mathematics.

A more topological view on 3-manifolds is presented in a set of notes by Hatcher: http://www.math.cornell.edu/~hatcher/3M/3Mfds.pdf

For mapping class groups, i.e. groups of homeomorphisms of surfaces, you may have a look at Farb's and Margalit's http://www.math.ethz.ch/~bgabi/Farb%20Magalit%20January%202011%20version.pdf . This is not directly related to the Geometrization Conjecture, but mapping class groups are extremely important both in 2-dimensional and 3-dimensional geometry. 

Edit: At some point you have also to learn some differential topology to understand geometric topology. I myself learned differential topology (partly) from the book by Bröcker and Jänich, but this is a little bit terse - there might be better choices. But the nice thing is that Thurston's book does not really presuppose any deeper knowledge in differential topology. 

I want also to comment than none of the above sources says anything about the proof of the geometrization conjecture; but I think, it would be unreasonable to try to understand the proof with your current background anyhow. 


  [1]: http://books.google.de/books?id=9kkuP3lsEFQC&printsec=frontcover&dq=Three-Dimensional+Geometry+and+Topology&hl=de&sa=X&ei=rml1UdjXH5SE0QHSs4GQBw&ved=0CDMQ6AEwAA#v=onepage&q=Three-Dimensional%2520Geometry%2520and%2520Topology&f=false