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 H{beta} emission line for a total of 44 quasars, and for the broad H{alpha} 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 H{beta} and 13 H{alpha} lags with JAVELIN, 42 H{beta} and 17 H{alpha} lags with CREAM, and 16 H{beta} and eight H{alpha} 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 H{beta}-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 H{alpha} emission is consistent with or slightly longer than that of H{beta}. 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 M_BH_-{sigma}* relationship, and are also consistent with single-epoch black hole mass measurements. This work increases the current sample size of reverberation-mapped active galaxies by about two- thirds and represents the first large sample of reverberation mapping observations beyond the local universe (z<0.3).