What would be the necessary background knowledge for understanding Poincaré's Conjecture

Question

I'm an interested graduate student from engineering. I'd like to study some geometry in my spare time. I know this sounds too ambitious but I always have an interest in geometry and would like to do this in long term even as a hobby. Just wondering what prerequisite textbooks would be required before one can understand advanced stuff such as Hamilton and Perelman work in Poincaré's conjecture, i.e. Ricci flows etc. My only related background is graduate real analysis (e.g. Folland) and abstract algebra (Dummit and Foote). Thank you.