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 1.5 to 10 m in diameter, located primarily in the Northern Hemisphere. We provide an overview of SAGUARO's telescopic resources, its pipeline for transient detection, and its database for candidate visualization. We describe SAGUARO's discovery component, which utilizes the 5deg2 field of view optical imager on the Mt. Lemmon 1.5 m telescope, reaching limits of ≈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/Virgo's 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 deg2 respectively, 17.6, 1.4, and 41.8 hr after the initial GW alerts. We covered 7.8%, 3.0%, and 5.1% of the total probability within the GW event localizations, reaching 3σ limits of 19.8, 21.3, and 20.8 AB mag, respectively. Although no viable counterparts associated with these events were found, we recovered six known transients and ruled out five 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.