In the Spring of 2011 we carried out a 2.5 month reverberation mapping campaign using the 3 m Shane telescope at Lick Observatory, monitoring 15 low-redshift Seyfert 1 galaxies. This paper describes the observations, reductions and measurements, and data products from the spectroscopic campaign. The reduced spectra were fitted with a multicomponent model in order to isolate the contributions of various continuum and emission-line components. We present light curves of broad emission lines and the active galactic nucleus (AGN) continuum, and measurements of the broad H beta line widths in mean and rms spectra. For the most highly variable AGNs we also measured broad Ha line widths and velocity centroids from the nightly spectra. In four AGNs exhibiting the highest variability amplitudes, we detect anticorrelations between broad H beta width and luminosity, demonstrating that the broad-line region "breathes" on short timescales of days to weeks in response to continuum variations. We also find that broad Ha velocity centroids can undergo substantial changes in response to continuum variations; in NGC 4593, the broad H beta velocity shifted by similar to 250 km s(-1) over a 1 month period. This reverberation-induced velocity shift effect is likely to contribute a significant source of confusion noise to binary black hole searches that use multi-epoch quasar spectroscopy to detect binary orbital motion. We also present results from simulations that examine biases that can occur in measurement of broad-line widths from rms spectra due to the contributions of continuum variations and photon-counting noise. ; National Science Foundation (NSF) AST-1108835, 1412693, 1107865, 1107812, 1412315, 1108665 ; NSF through the Graduate Research Fellowship Program ; NSF grant AST-1211916 ; Gary & Cynthia Bengier ; Richard & Rhoda Goldman Fund ; TABASGO Foundation ; Christopher R. Redlich Fund ; Packard Research Fellowship ; National Aeronautics and Space Administration (NASA) ; Gemini-CONICYT 32120009 ; NSF AST-1102845 ; National Research Foundation of Korea (NRF) grant - Korea government (MEST) 2012-006087 ; NSF Research at Undergraduate Institutions (RUI) grant AST-1312296 ; NSF Astronomy and Astrophysics Postdoctoral Fellowship AST-1302771 ; Grainger Foundation ; NASA ; Alfred P. Sloan Foundation ; NSF ; U.S. Department of Energy ; Japanese Monbukagakusho ; Max Planck Society ; Higher Education Funding Council for England ; Astronomy