While tidal disruption events (TDEs) have long been heralded as laboratories for the study of quiescent black holes, the small number of known TDEs and uncertainties in their emission mechanism have hindered progress towards this promise. Here present 17 new TDEs that have been detected recently by the Zwicky Transient Facility along with Swift UV and X-ray follow-up observations. Our homogeneous analysis of the optical/UV light curves, including 22 previously known TDEs from the literature, reveals a clean separation of light curve properties with spectroscopic class. The TDEs with Bowen fluorescence features in their optical spectra have smaller blackbody radii, as well as longer rise times and higher disruption rates compared to the rest of the sample. The Bowen fluorescence mechanism requires a high density which can be reached at smaller radii, which in turn yields longer diffusion timescales. Thus, the difference in rise times suggests the pre-peak TDE light curves are governed not by the fallback timescale, but instead by the diffusion of photons through the tidal debris. The small subset of TDEs that show only helium emission lines in their spectra have the longest rise times, the highest luminosities and the lowest rates. We also report, for the first time, the detection of soft X-ray flares from a TDE on day timescales. Based on the fact the flares peak at a luminosity similar to the optical/UV blackbody luminosity, we attribute them to brief glimpses through a reprocessing layer that otherwise obscures the inner accretion flow.