Which is the best book for studying geometric flows?

For mean curvature flow, to me the easiest one is Zhu's lectures on mean curvature flow. It covers the simplest cases (hypersurfaces) and the "classical" techniques/results, for example,

• De-Turck trick for the existence of the flow,
• Calculations of evolution equations of geometric quantities, use of maximum principle (scalar and matrix),
• Huisken's monotonicity formula, basic classification of singularities,
• A version of a well-known iteration technique in elliptic PDE

This book was published in 2002 so it definitely does not cover the whole topics well. But as a first read you can give it a go.

I recommend these books. Because these books are not too old and actually useful for poincare conjecture. Ricci flow of Poincare conjecture The book of Hamilton

In my opinion the easiest one for studying the Ricci flow is the book "The Ricci flow: An Introduction", written by Chow and Knoph.