We present branching fraction measurements for the decays B0→ρ0K*0, B0→f0K*0, and B0→ρ-K*+, where K* is an S-wave (Kπ)0* or a K*(892) meson; we also measure B0→f0K2*(1430)0. For the K*(892) channels, we report measurements of longitudinal polarization fractions (for ρ final states) and direct CP violation asymmetries. These results are obtained from a sample of (471.0±2.8)×106 BB̅ pairs collected with the BABAR detector at the PEP-II asymmetric-energy e+e- collider at the SLAC National Accelerator Laboratory. We observe ρ0K*(892)0, ρ0(Kπ)0*0, f0K*(892)0, and ρ-K*(892)+ with greater than 5σ significance, including systematics. We report first evidence for f0(Kπ)0*0 and f0K2*(1430)0, and place an upper limit on ρ-(Kπ)0*+. Our results in the K*(892) channels are consistent with no direct CP violation.