We study the coarse geometry of curve graphs and related graphs for connected, compact, orientable surfaces. We prove that the separating curve graph of a surface is a hierarchically hyperbolic space, as defined by Behrstock, Hagen and Sisto, whenever it is connected. It also automatically has the coarse median property defined by Bowditch. Consequences for the separating curve graph include a distance formula analogous to Masur and Minsky's distance formula for the mapping class group, an upper bound on the maximal dimension of quasiflats, and the existence of a quadratic isoperimetric inequality. We also describe surgery arguments for studying the coarse geometry of curve graphs and similar graphs. Specifically, we give a new proof of the uniform hyperbolicity of the curve graphs, extending methods of Przytycki and Sisto. We also give an elementary proof of Masur and Minsky's result that the disc graphs are quasi-convex in the curve graphs. Moreover, we show that the constant of quasiconvexity is independent of the surface, as also shown in work of Hamenstädt.