  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 ($\alpha_r = 2.01\pm0.02$), as expected in the
expanding fireball model. There are, however, individual SNe that are clearly
inconsistent with $\alpha_r=2$. 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 $\alpha=2$, 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.
