Early-time observations of Type Ia supernovae (SNe Ia) are essential to constrain the properties of their progenitors. In this paper, we present high-quality light curves of 127 SNe Ia discovered by the Zwicky Transient Facility (ZTF) in 2018. We describe our method to perform forced point-spread function photometry, which can be applied to other types of extragalactic transients. With a planned cadence of six observations per night (three g + three r), all of the 127 SNe Ia are detected in both g and r bands more than 10 days (in the rest frame) prior to the epoch of g-band maximum light. The redshifts of these objects range from z = 0.0181 to 0.165; the median redshift is 0.074. Among the 127 SNe, 50 are detected at least 14 days prior to maximum light (in the rest frame), with a subset of nine objects being detected more than 17 days before g-band peak. This is the largest sample of young SNe Ia collected to date; it can be used to study the shape and color evolution of the rising light curves in unprecedented detail. We discuss six peculiar events in this sample: one 02cx-like event ZTF18abclfee (SN 2018crl), one Ia-CSM SN ZTF18aaykjei (SN 2018cxk), and four objects with possible super-Chandrasekhar mass progenitors: ZTF18abhpgje (SN 2018eul), ZTF18abdpvnd (SN 2018dvf), ZTF18aawpcel (SN 2018cir), and ZTF18abddmrf (SN 2018dsx).