Astrometry can bring powerful constraints to bear on a variety of scientific questions about neutron stars, including their origins, astrophysics, evolution, and environments. Using phase-referenced observations at the VLBA, in conjunction with pulsar gating and in-beam calibration, we have measured the parallaxes and proper motions for 14 pulsars. The smallest measured parallax in our sample is 0.13+-0.02 mas for PSR B1541+09, which has a most probable distance of 7.2+1.3-1.1 kpc. We detail our methods, including initial VLA surveys to select candidates and find in-beam calibrators, VLBA phase-referencing, pulsar gating, calibration, and data reduction. The use of the bootstrap method to estimate astrometric uncertainties in the presence of unmodeled systematic errors is also described. Based on our new model-independent estimates for distance and transverse velocity, we investigate the kinematics and birth sites of the pulsars and revisit models of the Galactic electron density distribution. We find that young pulsars are moving away from the Galactic plane, as expected, and that age estimates from kinematics and pulsar spindown are generally in agreement, with certain notable exceptions. Given its present trajectory, the pulsar B2045-16 was plausibly born in the open cluster NGC 6604. For several high-latitude pulsars, the NE2001 electron density model underestimates the parallax distances by a factor of two, while in others the estimates agree with or are larger than the parallax distances, suggesting that the interstellar medium is irregular on relevant length scales. The VLBA astrometric results for the recycled pulsar J1713+0747 are consistent with two independent estimates from pulse timing, enabling a consistency check between the different reference frames. ; Comment: 16 pages, 9 figures, 4 tables; results unchanged; revised version accepted by ApJ