I want to set a course toward understanding Perelman's proof of the Geometrization Conjecture. I realize this will be a lengthy undertaking, but hopefully only on the order of one to two years.
I am currently studying John Lee's Introduction to Smooth Manifolds and Steven Weintraub's Fundamentals of Algebraic Topology. I will be taking a graduate course on 4-manifolds next semester.
Could you please suggest (1) which subject areas I should study toward this eventual goal and (2) any texts you particularly like?
I believe that the canonical reference is the book "Geometrisation of 3-manifolds" by Laurent Bessières, Gérard Besson, Michel Boileau, Sylvain Maillot and Joan Porti. It is freely available on Besson's web page: here.
The whole purpose of this book is to present a full detailed proof of geometrisation, based of course (partly) on Perelman's work, but made as comprehensible as possible. Of course, it may still require you to consult other references to understand it, but the book does provide many references.