The structure and kinematics of the Milky Way (MW) stellar halo provide a unique archaeological record of the MW's formation, past evolution, and accretion history. We present studies of the kinematic structure of the MW stellar halo with the Halo Assembly in Lambda-CDM: Observations in 7 Dimensions (HALO7D) survey. The HALO7D dataset consists of Keck II/DEIMOS spectroscopy and Hubble Space Telescope-measured proper motions of Milky Way (MW) halo main-sequence turnoff stars in the CANDELS fields. We first present the HALO7D pilot study, in which we use 13 distant main-sequence MW halo stars make the first estimate of the velocity anisotropy using 3D kinematic information outside of the solar neighborhood. Next, we present the spectroscopic component of the full HALO7D survey, and discuss target selection, observing strategy, and survey properties. We present a new method of measuring line-of-sight (LOS) velocities by combining multiple spectroscopic observations of a given star, utilizing Bayesian hierarchical modeling. We estimate the LOS velocity dispersions in the four fields and find that they are consistent with one another. We perform mock HALO7D surveys using the synthetic survey software Galaxia to ``observe'' the Bullock & Johnston (2005) accreted stellar halos. Based on these simulated datasets, the consistent LOS velocity distributions across the four HALO7D fields indicates that the HALO7D sample is dominated by stars from the same massive (or few relatively massive) accretion event(s). Finally, we present the proper motions for the HALO7D sample, and use the 3D kinematic measurements to estimate velocity anisotropy. We find that velocity anisotropy varies from field to field, which suggests that the halo is not phase mixed at r~23 kpc. We explore the anisotropy variation across the skies of two stellar halos from the Latte suite of FIRE-2 simulations, finding that both simulated galaxies show anisotropy variation over a similar range to the variation observed across the four HALO7D fields. The accretion histories of the two simulated galaxies result in different anisotropy variation patterns; spatially mapping velocity anisotropy is thus a way forward in characterizing the accretion history of the Galaxy.