The discovery of the electromagnetic counterpart to the binary neutron star (NS) merger GW170817 has opened the era of gravitational- wave multimessenger astronomy. Rapid identification of the optical/infrared kilonova enabled a precise localization of the source, which paved the way to deep multiwavelength follow-up and its myriad of related science results. Fully exploiting this new territory of exploration requires the acquisition of electromagnetic data from samples of NS mergers and other gravitational-wave sources. After GW170817, the frontier is now to map the diversity of kilonova properties and provide more stringent constraints on the Hubble constant, and enable new tests of fundamental physics. The Vera C. Rubin Observatory's Legacy Survey of Space and Time can play a key role in this field in the 2020s, when an improved network of gravitational- wave detectors is expected to reach a sensitivity that will enable the discovery of a high rate of merger events involving NSs (åisebox-0.5ex tens per year) out to distances of several hundred megaparsecs. We design comprehensive target-of-opportunity observing strategies for follow-up of gravitational-wave triggers that will make the Rubin Observatory the premier instrument for discovery and early characterization of NS and other compact-object mergers, and yet unknown classes of gravitational-wave events.