We present a Hubble Space Telescope (HST) study of stellar proper motions (PMs) for four fields spanning 200 degrees along the Sagittarius (Sgr) stream: one field in the trailing arm, one field near the Sgr dSph tidal radius, and two fields in the leading arm. From data with 6-9 year time baselines, we determine absolute PMs of dozens of individual stars per field, using established techniques that use distant background galaxies to define a stationary reference frame. Stream stars are identified based on combined color-magnitude diagram (CMD) and PM information. The results are broadly consistent with the few existing PM measurements for the Sgr dwarf spheroidal galaxy (dSph) and the trailing arm. However, our new results provide the highest PM accuracy for the stream to date, the first PM measurements for the leading arm, and the first PM measurements for individual stream stars [We also serendipitously determine the PM of the globular cluster NGC 6652 to be (\(\mu_{\rm W}\), \(\mu_{\rm N}\)) = (5.66 \(\pm\) 0.07, \(-\)4.45 \(\pm\) 0.10) mas yr\(^{-1}\)]. In the trailing-arm field, the individual PMs allow us to kinematically separate trailing-arm stars from leading-arm stars that are 360 degrees further ahead in their orbit around the Milky Way (MW). Also, in three of our fields we find indications that two distinct kinematical components may exist within the same arm and wrap of the stream. Qualitative comparison of the HST data to the predictions of the Law & Majewski N-body model of the stream show that the PM measurements closely follow the predicted trend with Sgr longitude. While this does not necessarily indicate that the triaxial MW dark halo shape inferred from the \(N\)-body model is correct, it does provide a successful consistency check using PM data that the model was not tailored to reproduce. Quantitative data-model comparisons will be presented in a companion paper.