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While it is clear that Type Ia supernovae (SNe) are the result of thermonuclear explosions in C/O white dwarfs (WDs), a great deal remains uncertain about the binary companion that facilitates the explosive disruption of the WD. Here, we present a comprehensive analysis of a large, unique data set of 127 SNe$\,$Ia with exquisite coverage by the Zwicky Transient Facility (ZTF). High-cadence (six observations per night) ZTF observations allow us to measure the SN rise time and examine its initial evolution. We develop a Bayesian framework to model the early rise as a power law in time, which enables the inclusion of priors in our model. For a volume-limited subset of normal SNe$\,$Ia, we find that the mean power-law index is consistent with 2 in the $r_\mathrm{ZTF}$-band ($α_r = 2.01\pm0.02$), as expected in the expanding fireball model. There are, however, individual SNe that are clearly inconsistent with $α_r=2$. We estimate a mean rise time of 18.9$\,$d (with a range extending from $\sim$15 to 22$\,$d), though this is subject to the adopted prior. We identify an important, previously unknown, bias whereby the rise times for higher-redshift SNe within a flux-limited survey are systematically underestimated. This effect can be partially alleviated if the power-law index is fixed to $α=2$, in which case we estimate a mean rise time of 21.7$\,$d (with a range from $\sim$18 to 23$\,$d). The sample includes a handful of rare and peculiar SNe$\,$Ia. Finally, we conclude with a discussion of lessons learned from the ZTF sample that can eventually be applied to observations from the Vera C. Rubin Observatory.