We describe the Gems of the Galaxy Zoos (Zoo Gems) project, a gap-filler project using short windows in the Hubble Space Telescope's schedule. As with previous snapshot programs, targets are taken from a pool based on position; we combine objects selected by volunteers in both the Galaxy Zoo and Radio Galaxy Zoo citizen-science projects. Zoo Gems uses exposures with the Advanced Camera for Surveys to address a broad range of topics in galaxy morphology, interstellar-medium content, host galaxies of active galactic nuclei, and galaxy evolution. Science cases include studying galaxy interactions, backlit dust in galaxies, post-starburst systems, rings and peculiar spiral patterns, outliers from the usual color-morphology relation, Green Pea compact starburst systems, double radio sources with spiral host galaxies, and extended emission-line regions around active galactic nuclei. For many of these science categories, final selection of targets from a larger list used public input via a voting process. Highlights to date include the prevalence of tightly wound spiral structure in blue, apparently early-type galaxies, a nearly complete Einstein ring from a group lens, redder components at lower surface brightness surrounding compact Green Pea starbursts, and high-probability examples of spiral galaxies hosting large double radio sources.