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 toward this promise. Here we 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, lower optical luminosities, and higher disruption rates compared to the rest of the sample. 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. A high detection rate of Bowen lines in TDEs with small photometric radii could be explained by the high density that is required for this fluorescence mechanism. The stellar debris can provide a source for this dense material. Diffusion of photons through this debris may explain why the rise and fade timescale of the TDEs in our sample are not correlated. We also report, for the first time, the detection of soft X-ray flares from a TDE on similar to day timescales. Based on the fact that the X-ray 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.