Using the Very Large Array, we have measured the proper motions of 28 radio pulsars. On average, the pulsars studied are fainter and more distant than those studied in earlier work, reducing the selection biases inherent in surveys restricted to the solar neighborhood. The typical measurement precision achieved is a few milliarcseconds per year, corresponding to a few tens of kilometers per second for a pulsar a kiloparsec away. While our results compare well with higher precision measurements done using very long baseline interferometry, we find that several earlier proper-motion surveys appear to have reported overly optimistic measurement uncertainties, most likely because of a failure to fully account for ionospheric effects. We discuss difficulties inherent in estimating pulsar velocities from proper motions given poorly constrained pulsar distances. Our observations favor a distribution with 20% of pulsars in a low-velocity component (σrm1D=99 km/s) and 80% in a high-velocity component (σ1D=294 km/s). Furthermore, our sample is consistent with a scale height of pulsar birthplaces comparable to the scale height of the massive stars that are their presumed progenitors. No evidence is found in our data for a significant population of young pulsars born far from the plane. We find that estimates of pulsar ages based on kinematics agree well with the canonical spin-down age estimate, but agreement is improved if braking indexes are drawn from a Gaussian distribution centered at n=3 with width 0.8.