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One of the keys to the success of multimessenger astronomy is the rapididentification of the electromagnetic wave counterpart, kilonova (KN), of thegravitational-wave (GW) event. Despite its importance, it is hard to find a KNassociated with a GW event, due to a poorly constrained GW localization map andnumerous signals that could be confused as a KN. Here, we present theGravitational-wave Electromagnetic wave Counterpart Korean Observatory (GECKO)project, the GECKO observation of GW190425, and prospects of GECKO in thefourth observing run (O4) of the GW detectors. We outline our follow-upobservation strategies during O3. In particular, we describe ourgalaxy-targeted observation criteria that prioritize based on galaxyproperties. Armed with this strategy, we performed an optical and/ornear-infrared follow-up observation of GW190425, the first binary neutron starmerger event during the O3 run. Despite a vast localization area of 7460 deg^2,we observed 621 host galaxy candidates, corresponding to 29.5% of the scores weassigned, with most of them observed within the first 3 days of the GW event.Ten transients were discovered during this search, including a new transientwith a host galaxy. No plausible KN was found, but we were still able toconstrain the properties of potential KNe using upper limits. The GECKOobservation demonstrates that GECKO can possibly uncover a GW170817-like KN ata distance less than 200 Mpc if the localization area is of the order ofhundreds of square degrees, providing a bright prospect for the identificationof GW electromagnetic wave counterparts during the O4 run.

Gravitational-wave Electromagnetic Counterpart Korean Observatory (GECKO): GECKO Follow-up Observation of GW190425