We present Searches After Gravitational-waves Using ARizona Observatories (SAGUARO), a comprehensive effort dedicated to the discovery and characterization of optical counterparts to gravitational wave (GW) events. SAGUARO utilizes ground-based facilities ranging from 1.5m to 10m in diameter, located primarily in the Northern Hemisphere. We provide an overview of SAGUAROs telescopic resources, pipeline for transient detection, and database for candidate visualization. We describe SAGUAROs discovery component, which utilizes the $5$~deg$^2$ field-of-view optical imager on the Mt. Lemmon 1.5m telescope, reaching limits of $approx 21.3$~AB mag while rapidly tiling large areas. We also describe the follow-up component of SAGUARO, used for rapid vetting and monitoring of optical candidates. With the onset of Advanced LIGO/Virgos third observing run, we present results from the first three SAGUARO searches following the GW events S190408an, S190425z and S190426c, which serve as a valuable proof-of-concept of SAGUARO. We triggered and searched 15, 60 and 60 deg$^{2}$ respectively, 17.6, 1.4 and 41.8 hrs after the initial GW alerts. We covered 7.8, 3.0 and 5.1% of the total probability within the GW event localizations, reaching 3$sigma$ limits of 19.8, 21.3 and 20.8 AB mag, respectively. Although no viable counterparts associated with these events were found, we recovered 6 known transients and ruled out 5 potential candidates. We also present Large Binocular Telescope spectroscopy of PS19eq/SN2019ebq, a promising kilonova candidate that was later determined to be a supernova. With the ability to tile large areas and conduct detailed follow-up, SAGUARO represents a significant addition to GW counterpart searches.
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