We report searches for B-meson decays to the charmless final states rhoK* and f0(980)K* with a sample of 232x10(6) BB pairs collected with the BABAR detector at the PEP-II e+e- collider. We measure in units of 10(-6) the following branching fractions, where the first error quoted is statistical and the second systematic, or upper limits are given at the 90% confidence level: B(B+-->rho0K*+)<6.1, B(B+-->rho+K*0)=9.6+/-1.7+/-1.5, B(B0-->rho-K*+)<12.0, B(B0-->rho0K*0)=5.6+/-0.9+/-1.3, B(B+-->f0(980)K*+)=5.2+/-1.2+/-0.5, and B(B0-->f0(980)K*0)<4.3. For the significant modes, we also measure the fraction of longitudinal polarization and the charge asymmetry: fL(B+-->rho+K*0)=0.52+/-0.10+/-0.04, fL(B0-->rho0K*0)=0.57+/-0.09+/-0.08, ACP(B+-->rho+K*0)=-0.01+/-0.16+/-0.02, ACP(B0-->rho0K*0)=0.09+/-0.19+/-0.02, and ACP(B+-->f0(980)K*+)=-0.34+/-0.21+/-0.03.