The discovery of the electromagnetic counterparts to the binary neutron star merger GW170817 has opened the era of GW+EM multi-messenger astronomy. Exploiting this breakthrough requires increasing samples to explore the diversity of kilonova behaviour and provide more stringent constraints on the Hubble constant, and tests of fundamental physics. LSST can play a key role in this field in the 2020s, when the gravitational wave detector network is expected to detect higher rates of merger events involving neutron stars ($\sim$10s per year) out to distances of several hundred Mpc. Here we propose comprehensive target-of-opportunity (ToOs) strategies for follow-up of gravitational-wave sources that will make LSST the premiere machine for discovery and early characterization for neutron star mergers and other gravitational-wave sources.