To understand Perelman's proof of the Poincaré Conjecture, you need a solid background in Riemannian geometry. Many books can be used for an introduction to this field. There are two books I like on this subject : Riemannian Geometry, by Gallot, Hulin and Lafontaine and Riemannian Geometry by Petersen.

After, you can try to learn about Ricci flow, a good starting point is Chow and Knopff's "The Ricci Flow : an Introduction". It covers the basics of Ricci flow including Hamilton's theorem that on a compact 3-manifold with $Ric>0$, the (normalized) flow will converge to constant curvature.

Then, if you want to go into Perelman's work, there is the book "Ricci Flow and the Poincaré Conhecture" by Morgan and Tian. However you also have to understand Thurston's Geommetrization Conjecture, so you need a solid background in 3-manifold topology,
I don't know the references for this part, maybe Thurston's lecture notes ?

Another interesting road is to study the proof of the differentiable sphere theorem by Brendle and Schoen, a good reference is Brendle's "Ricci Flow and the Sphere Theorem".

I Hope that was helpful.