We present the most extensive combined photometric and spectroscopic study to date of the enormous globular cluster (GC) system around M87, the central giant elliptical galaxy in the nearby Virgo Cluster. Using observations from DEIMOS and the Low Resolution Imaging Spectrometer at Keck, and Hectospec on the Multiple Mirror Telescope, we derive new, precise radial velocities for 451 GCs around M87, with projected radii from ~5 to 185 kpc. We combine these measurements with literature data for a total sample of 737 objects, which we use for a re-examination of the kinematics of the GC system of M87. The velocities are analyzed in the context of archival wide-field photometry and a novel Hubble Space Telescope catalog of half-light radii, which includes sizes for 344 spectroscopically confirmed clusters. We use this unique catalog to identify 18 new candidate ultracompact dwarfs and to help clarify the relationship between these objects and true GCs. We find much lower values for the outer velocity dispersion and rotation of the GC system than in earlier papers and also differ from previous work in seeing no evidence for a transition in the inner halo to a potential dominated by the Virgo Cluster, nor for a truncation of the stellar halo. We find little kinematical evidence for an intergalactic GC population. Aided by the precision of the new velocity measurements, we see significant evidence for kinematical substructure over a wide range of radii, indicating that M87 is in active assembly. A simple, scale-free analysis finds less dark matter within ~85 kpc than in other recent work, reducing the tension between X-ray and optical results. In general, out to a projected radius of ~150 kpc, our data are consistent with the notion that M87 is not dynamically coupled to the Virgo Cluster; the core of Virgo may be in the earliest stages of assembly.