The 'Spectroscopy and H-band Imaging of Virgo cluster galaxies' (SHIVir) survey is an optical and near-infrared survey which combines SDSS photometry, deep H-band photometry, and long-slit optical spectroscopy for 190 Virgo cluster galaxies covering all morphological types over the stellar mass range log (M*/M_{sun}_) = 7.8-11.5. We present the spectroscopic sample selection, data reduction, and analysis for this SHIVir sample. We have used and optimized the pPXF routine to extract stellar kinematics from our data. Ultimately, resolved kinematic profiles (rotation curves and velocity dispersion profiles) are available for 133 SHIVir galaxies. A comprehensive data base of photometric and kinematic parameters for the SHIVir sample is presented with grizH magnitudes, effective surface brightnesses, effective and isophotal radii, rotational velocities, velocity dispersions, and stellar and dynamical masses. Parameter distributions highlight some bimodal distributions and possible sample biases. A qualitative study of resolved extended velocity dispersion profiles suggests a link between the so-called 'sigma-drop' kinematic profile and the presence of rings in lenticular S0 galaxies. Rising dispersion profiles are linked to early- type spirals or dwarf ellipticals for which a rotational component is significant, whereas peaked profiles are tied to featureless giant ellipticals.