The ROSAT All-Sky Survey (RASS) was the first imaging X-ray survey of the entire sky. Combining the RASS Bright and Faint Source Catalogs (Cat. <IX/10>, 1RXS and <IX/29>) yields an average of about three X-ray sources per square degree. However, while X-ray source counterparts are known to range from distant quasars to nearby M dwarfs, the RASS data alone are often insufficient to determine the nature of an X-ray source. As a result, large-scale follow-up programs are required to construct samples of known X-ray emitters. We use optical data produced by the Sloan Digital Sky Survey (SDSS) to identify 709 stellar X-ray emitters cataloged in the RASS and falling within the SDSS Data Release 1 footprint. Most of these are bright stars with coronal X-ray emission unsuitable for SDSS spectroscopy, which is designed for fainter objects (g>15[mag]). Instead, we use SDSS photometry, correlations with the Two Micron All Sky Survey and other catalogs, and spectroscopy from the Apache Point Observatory 3.5m telescope to identify these stellar X-ray counterparts. Our sample of 707 X-ray-emitting F, G, K, and M stars is one of the largest X-ray-selected samples of such stars. We identify 17 new X-ray-emitting DA (hydrogen) WDs, of which three are newly identified WDs. We report on follow-up observations of three candidate cool X-ray-emitting WDs (one DA and two DB (helium) WDs); we have not confirmed X-ray emission from these WDs.