We present a high-precision radial velocity (RV) survey of 719 FGKM stars, which host 164 known exoplanets and 14 newly discovered or revised exoplanets and substellar companions. This catalog updated the orbital parameters of known exoplanets and long-period candidates, some of which have decades-longer observational baselines than they did upon initial detection. The newly discovered exoplanets range from warm sub-Neptunes and super-Earths to cold gas giants. We present the catalog sample selection criteria, as well as over 100,000 RV measurements, which come from the Keck-HIRES, APF-Levy, and Lick- Hamilton spectrographs. We introduce the new RV search pipeline RVSearch (http://california-planet-search.github.io/rvsearch/) that we used to generate our planet catalog, and we make it available to the public as an open-source Python package. This paper is the first study in a planned series that will measure exoplanet occurrence rates and compare exoplanet populations, including studies of giant planet occurrence beyond the water ice line, and eccentricity distributions to explore giant planet formation pathways. We have made public all radial velocities and associated data that we use in this catalog.