We describe a 20 year survey carried out by the Lick-Carnegie Exoplanet Survey Team (LCES), using precision radial velocities from HIRES on the Keck I telescope to find and characterize extrasolar planetary systems orbiting nearby F, G, K, and M dwarf stars. We provide here 60,949 precision radial velocities for 1624 stars contained in that survey. We tabulate a list of 357 significant periodic signals that are of constant period and phase, and not coincident in period and/or phase with stellar activity indices. These signals are thus strongly suggestive of barycentric reflex motion of the star induced by one or more candidate exoplanets in Keplerian motion about the host star. Of these signals, 225 have already been published as planet claims, 60 are classified as significant unpublished planet candidates that await photometric follow-up to rule out activity-related causes, and 54 are also unpublished, but are classified as "significant" signals that require confirmation by additional data before rising to classification as planet candidates. Of particular interest is our detection of a candidate planet with Msin(i) = 3.8M(circle plus), and P = 9.9 days orbiting Lalande 21185, the fourth-closest main-sequence star to the Sun. For each of our exoplanetary candidate signals, we provide the period and semi-amplitude of the Keplerian orbital fit, and a likelihood ratio estimate of its statistical significance. We also tabulate 18 Keplerian-like signals that we classify as likely arising from stellar activity.