The PHANGS program is building the first dataset to enable the multi-phase, multi-scale study of star formation across the nearby spiral galaxy population. This effort is enabled by large Treasury programs with ALMA, VLT/MUSE, and HST, with which we have obtained CO(2-1) imaging, optical spectroscopic mapping, and high resolution UV-optical imaging, respectively. Here, we present PHANGS-HST, which is obtaining five band NUV-U-B-V-I imaging of the disks of 38 spiral galaxies at distances of 4-23 Mpc, and parallel V and I band imaging of their halos, to provide a census of tens of thousands of compact star clusters and associations. The combination of HST, ALMA, and VLT/MUSE observations will yield an unprecedented joint catalog of the observed and physical properties of ∼100,000 star clusters, associations, HII regions, and molecular clouds. With these basic units of star formation, PHANGS will systematically chart the evolutionary cycling between gas and stars, across a diversity of galactic environments found in nearby galaxies. We discuss the design of the PHANGS-HST survey, and provide an overview of the HST data processing pipeline and first results, highlighting new methods for selecting star cluster candidates, morphological classification of candidates with convolutional neural networks, and identification of stellar associations over a range of physical scales with a watershed algorithm. We describe the cross-observatory imaging, catalogs, and software products being released at MAST, which will seed a broad range of community science, in particular, upcoming JWST study of dust embedded star formation and ISM physics.