On 1999 May 26, one of the Sloan Digital Sky Survey (SDSS) fiber-fed spectrographs saw astronomical first light. This was followed by the first spectroscopic commissioning run during the dark period of 1999 June. We present here the first hour of extragalactic spectroscopy taken during these early commissioning stages: an observation of the Coma cluster of galaxies. Our data samples the southern part of this cluster, out to a radius of 15 (1.8 h-1 Mpc, approximately to the virial radius) and thus fully covers the NGC 4839 group. We outline in this paper the main characteristics of the SDSS spectroscopic systems and provide redshifts and spectral classifications for 196 Coma galaxies, of which 45 redshifts are new. For the 151 galaxies in common with the literature, we find excellent agreement between our redshift determinations and the published values, e.g., for the largest homogeneous sample of galaxies in common (63 galaxies observed by Colless & Dunn) we find a mean offset of 3 km s -1 and an rms scatter of only 24 km s -1. As part of our analysis, we have investigated four different spectral classification algorithms: measurements of the spectral line strengths, a principal component decomposition, a wavelet analysis and the fitting of spectral synthesis models to the data. We find that these classification schemes are in broad agreement and can provide physical insight into the evolutionary histories of our cluster galaxies. We find that a significant fraction (25%) of our observed Coma galaxies show signs of recent star formation activity and that the velocity dispersion of these active galaxies (emission-line and poststarburst galaxies) is 30% larger than the absorption-line galaxies. We also find no active galaxies within the central (projected) 200 h-1 kpc of the cluster. The spatial distribution of our Coma active galaxies is consistent with that found at higher redshift for the CNOC1 cluster survey. Beyond the core region, the fraction of bright active galaxies appears to rise slowly out to the virial radius and are randomly distributed within the cluster with no apparent correlation with the potential merger or postmerger of the NGC 4839 group. We briefly discuss possible origins of this recent galaxy star formation.