In this first paper in a series, we present 1298 low-redshift (z łesssim 0.2) optical spectra of 582 Type Ia supernovae (SNe Ia) observed from 1989 to 2008 as part of the Berkeley Supernova Ia Program (BSNIP). 584 spectra of 199 SNe Ia have well-calibrated light curves with measured distance moduli, and many of the spectra have been corrected for host-galaxy contamination. Most of the data were obtained using the Kast double spectrograph mounted on the Shane 3 m telescope at Lick Observatory and have a typical wavelength range of 3300-10 400 ̊A, roughly twice as wide as spectra from most previously published data sets. We present our observing and reduction procedures, and we describe the resulting SN Database, which will be an online, public, searchable data base containing all of our fully reduced spectra and companion photometry. In addition, we discuss our spectral classification scheme (using the SuperNova IDentification code, SNID; Blondin & Tonry), utilizing our newly constructed set of SNID spectral templates. These templates allow us to accurately classify our entire data set, and by doing so we are able to reclassify a handful of objects as bona fide SNe Ia and a few other objects as members of some of the peculiar SN Ia subtypes. In fact, our data set includes spectra of nearly 90 spectroscopically peculiar SNe Ia. We also present spectroscopic host-galaxy redshifts of some SNe Ia where these values were previously unknown. The sheer size of the BSNIP data set and the consistency of our observation and reduction methods make this sample unique among all other published SN Ia data sets and complementary in many ways to the large, low-redshift SN Ia spectra presented by Matheson et al. and Blondin et al. In other BSNIP papers in this series, we use these data to examine the relationships between spectroscopic characteristics and various observables such as photometric and host-galaxy properties.