The direct detection of gravitational waves (GWs) from merging binary black holes and neutron stars mark the beginning of a new era in gravitational physics, and it brings forth new opportunities to test theories of gravity. The properties of GW propagation are modified in alternative theories of gravity and are crucial observables to test gravity at cosmological distance. From the coincident detection of GW170817/GRB170817A, the propagation speed has already been measured so precisely that a class of gravity theories in which GWs propagate with the speed different from c has been ruled out. On the other hand, another conspicuous feature in modified gravity is the time variation of gravitational constant (G), which affects the amplitude damping of GWs. The constraint on the time variation of G for GWs has obtained from GW170817/GRB170817A but is still too weak. To deal with the current situation of testing gravity with GWs, we proposed a new universal framework for testing gravity with GWs, based on the generalized propagation of a GW in an effective field theory that describes modification of gravity at cosmological scales. Then we performed a parameter estimation study, showing that the future observation of GW can constrain the parameter of the amplitude damping in the generalized models of GW propagation down to a level of 1%. We also studied the time variation of the gravitational couplings in Horndeski theory by performing Monte Carlo-based numerical simulations. From the simulation results, we find that the current accelerating Universe prefers the models with less damping of GWs and that almost all models produced in the numerical simulation can be tested by the future GW observation. In this article, we review GW propagation test of gravity and its implication for cosmology, particularly focusing on our recent works.