I roughly read the Topping's LECTURES ON THE RICCI FLOW. There does not seem to be an introduction on surgery. Seemly, it is enough to deal singularity by blow up. Then, in order to know surgery, I read the Perelman's Ricci flow with surgery on three-manifold. But it is not easy to read. Are there any books that introduce the surgery, and likely to Topping's book (I feel Topping's book is friendly to read) ?
A group of five mathematicians (Bessières, Besson, Boileau, Maillot and Porti) have written a book aiming to give a complete proof of the Geometrisation Conjecture from pre-Perelman's results.
As such, they present a version of Ricci flow with surgery. I hope the book will appear suitable to you (I haven't read it in detail, but the introduction and first chapters seem remarkably clear to me).