We present reverberation mapping results from the first year of combined spectroscopic and photometric observations of the Sloan Digital Sky Survey Reverberation Mapping Project. We successfully recover reverberation time delays between the g+i band emission and the broad Hβ emission line for a total of 44 quasars, and for the broad Hα emission line in 18 quasars. Time delays are computed using the JAVELIN and CREAM software and the traditional interpolated cross-correlation function (ICCF): using well-defined criteria, we report measurements of 32 Hβ and 13 Hα lags with JAVELIN, 42 Hβ and 17 Hα lags with CREAM, and 16 Hβ and eight Hα lags with the ICCF. Lag values are generally consistent among the three methods, though we typically measure smaller uncertainties with JAVELIN and CREAM than with the ICCF, given the more physically motivated light curve interpolation and more robust statistical modeling of the former two methods. The median redshift of our Hβ-detected sample of quasars is 0.53, significantly higher than that of the previous reverberation mapping sample. We find that in most objects, the time delay of the Hα emission is consistent with or slightly longer than that of Hβ. We measure black hole masses using our measured time delays and line widths for these quasars. These black hole mass measurements are mostly consistent with expectations based on the local