A phase-space diagram of a cluster can be a simple plot of clustocentric velocity versus clustocentric radius for each member of a cluster. Using a suite of cosmological hydrodynamical simulations of clusters, we investigate where simulated galaxies fall in phase-space. We find that, by comparing populations of galaxies in different regions of phase-space, we can systematically change the mean infall time of that population into their host. This is very useful for understanding environmental effects as the presence of recent infallers mixed in with ancient infallers tends to camouflage the true environmental impact, and we are able to clearly reveal it using phase-space. We then apply this tool to observations of galaxies in groups and clusters to study gas stripping, jelly-fish galaxies and tidal-mass loss. We also consider a large sample of SDSS satellite galaxies and study how the time they spend in their host impacts on luminosity-weighted ages, specific star formation rates, metallicity and alpha-abundances of the satellites. Using phase-space, we reveal clear signatures that even low-mass hosts can impact the stellar mass growth of their massive satellites, and see tentative hints of the effect of protoclusters and their surroundings on initial galaxy growth as early as redshift three. In short, we demonstrate how phase-space diagrams are essential tools for understanding environmental effects acting on satellite galaxies.