Readings

For hyperbolic geometry, you can start by reading  some  of 1. and then some of 2. (and repeat 1,2,1,2...)


For topology in dimension three (which depends very much on hyperbolic geometry), start reading 5.  Afterward (possibly much later depending on your knowledge of topology) read some of 3. then some of 4.; and again (3,4,3,4..) back and forth.
 

1.  Hyperbolic geometry, the first 150 years , by John Milnor
2. Hyperbolic manifolds according to Thurston and Jorgensen by Michael Gromov.
3. Towards the Poincaré Conjecture and the Classification of 3-Manifolds,  by John Milnor
4.  Recent progress on the Poincare conjecture and classification of 3-manifolds.by John Morgan
5. "The Poincare conjecture" by John Milnor, and the "abstract". 
6. A lecture about the Poincare conjecture by Curt McMullen