This paper presents spectroscopy of supernovae (SNe) discovered in the first season of the Sloan Digital Sky Survey-II SN Survey. This program searches for and measures multi-band light curves of SNe in the redshift range z = 0.05-0.4, complementing existing surveys at lower and higher redshifts. Our goal is to better characterize the SN population, with a particular focus on SNe Ia, improving their utility as cosmological distance indicators and as probes of dark energy. Our SN spectroscopy program features rapid-response observations using telescopes of a range of apertures, and provides confirmation of the SN and host-galaxy types as well as precise redshifts. We describe here the target identification and prioritization, data reduction, redshift measurement, and classification of 129 SNe Ia, 16 spectroscopically probable SNe Ia, 7 SNe Ib/c, and 11 SNe II from the first season. We also describe our efforts to measure and remove the substantial host-galaxy contamination existing in the majority of our SN spectra.