Renormalized volume was introduced by Graham and Witten for conformally compact Einstein manifolds. Krasnov-Schlenker studied it for convex compact hyperbolic 3-manifolds and proved many of its fundamental properties. In a series of papers, we consider the Weil-Petersson gradient flow of the renormalized volume function $V_R$ on the space of convex compact structures $CC(N)$ of a 3-manifold $N$. For $\partial N$ incompressible, we use the flow to show that the infimum of  $V_R$ on $CC(N)$ is the Gromov norm of $N$. For $N$ acylindrical, we introduce a surgered flow and show that starting at any $M \in CC(N)$  the flow limits to $M_{geod}$, the unique manifold with totally geodesic convex core boundary. Using this, we prove a lower bound on the convex core volume function $V_C$ on $CC(N)$ given by $V_C(M)-V_C(M_geod) \geq Kd_WP(\partial M,\partial M_geod) - C$.