The transport properties of exciton-polaritons are studied with allowance for their polarization. Both classical multiple scattering effects and quantum effects such as weak localization are taken into account in the framework of a generalized kinetic equation. The longitudinal-transverse (TE-TM) splitting of polariton states which plays a role that is analogous to the spin-orbit splitting in electron systems is taken into account. The developed formalism is applied to calculate the particle and spin diffusion coefficients of exciton-polaritons, spin relaxation rates, and the polarization conversion efficiency under the conditions of the optical spin Hall effect. In contrast to the electron systems, a strong spin splitting does not lead to the antilocalization behavior of the particle diffusion coefficient, while quantum corrections to spin diffusion and polarization conversion can be both negative and positive depending on the spin splitting value.