We present the first study of high-precision internal proper motions (PMs) in a large sample of globular clusters, based on Hubble Space Telescope (HST) data obtained over the past decade with the ACS/WFC, ACS/HRC, and WFC3/UVIS instruments. We determine PMs for over 1.3 million stars in the central regions of 22 clusters, with a median number of similar to 60,000 stars per cluster. These PMs have the potential to significantly advance our understanding of the internal kinematics of globular clusters by extending past line-of-sight (LOS) velocity measurements to two or three-dimensional velocities, lower stellar masses, and larger sample sizes. We describe the reduction pipeline that we developed to derive homogeneous PMs from the very heterogeneous archival data. We demonstrate the quality of the measurements through extensive Monte Carlo simulations. We also discuss the PM errors introduced by various systematic effects and the techniques that we have developed to correct or remove them to the extent possible. We provide in electronic form the catalog for NGC 7078 (M 15), which consists of 77,837 stars in the central 2'.4. We validate the catalog by comparison with existing PM measurements and LOS velocities and use it to study the dependence of the velocity dispersion on radius, stellar magnitude (or mass) along the main sequence, and direction in the plane of the sky (radial or tangential). Subsequent papers in this series will explore a range of applications in globular-cluster science and will also present the PM catalogs for the other sample clusters.