The Neil Gehrels Swift Observatory has proven to be an extraordinary supernova (SN) observatory. The clearest application of Swift’s unique strengths is obtaining very early UV and X-ray data of young SNe, which enables robust constraints on their progenitor systems. As part of a year-long Swift Guest Investigator Key Project, we initiated a follow-up program to rapidly observe all of the nearest (distance < 35 Mpc or roughly z < 0.008) extragalactic transients without waiting for them to be spectroscopically classified as supernovae. Among the possible results were to measure any UV-bright radiative cooling following the shock breakout from core-collapse SNe and shock emission from the interaction of thermonuclear Type Ia SNe with a non-degenerate companion. Just as importantly, uniformly following up and analyzing a significant sample can constrain the fraction of events for which the shock emission is not seen. Here we present the UV and X-ray measurements performed during our campaign. Our sample of 24 observed triggers included three SNe Ia, six SNe II, three stripped-envelope, core-collapse SNe, five galactic transients, three extragalactic SN imposters, and four unconfirmed transients. For our sample, the median delay time from the discovery image to the first Swift image was 1.45 days. We tabulate the X-ray upper limits and find they are sufficiently deep to have detected objects as X-ray luminous as GRB060218/SN2006aj. Other X-ray-detected SNe such as SNe 2006bp, 2008D, and 2011dh would have been detectable in some of the observations. We highlight the spectroscopically classified Type II SN 2018hna with UV-optical light curves indicating a luminosity and flux evolution very similar to SN 1987A.