Novel numerical simulations investigating the Richtmyer-Meshkov instability (RMI) with surface tension are presented. We solve the two-phase compressible Euler equation with surface tension and interface reconstruction by a volume-of-fluid method. We validate and bridge existing theoretical models of surface tension's effects on the RMI in linear, transitional and nonlinear post-shock growth regimes. Under an appropriately constructed dimensional framework, we find good agreement with existing linear incompressible theory in the small-amplitude (linear) oscillatory regime for positive Atwood numbers, and we show that negative Atwood numbers can be accommodated by an appropriate modification to the theory. Next, we show good agreement with nonlinear theory for asymptotic interface growth in the limit of small surface tension. Finally, we heuristically identify a criterion for transition from the linear into the nonlinear oscillation regime. These results highlight the utility of this numerical method for compressible problems featuring surface tension, and pave the way for a broader investigation into mixed compressible/incompressible problems.