In this first paper in a series we present 1298 low-redshift (z≤0.2) optical spectra of 582 Type Ia supernovae (SNe Ia) observed from 1989 through 2008 as part of the Berkeley SN 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 Ang., roughly twice as wide as spectra from most previously published datasets. We present our observing and reduction procedures, and we describe the resulting SN Database (SNDB), which will be an online, public, searchable database 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 2007), utilising our newly constructed set of SNID spectral templates. These templates allow us to accurately classify our entire dataset, 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 dataset 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. [Abridged] ; Comment: 34 pages, 11 figures, 11 tables, revised version, re-submitted to MNRAS. Spectra will be released in January 2013. The SN Database homepage (http://hercules.berkeley.edu/database/index_public.html) contains the full tables, plots of all spectra, and our new SNID templates