I roughly read the Topping's LECTURES ON THE RICCI FLOW. Seemly, there is not introduction about surgery. Seemly,it is enough to deal singularity by blow up. Then, for knowing surgery I read the Perelman's Ricci flow with surgery on three-manifold. But it is not easy to read. Are there any book introduct 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).