The premise of "near-field cosmology" is that the study of nearby low-mass galaxies on a star-by-star basis has implications that extend far beyond the local Universe and include the nature of dark matter, lives and deaths of the first stars, cosmic reionization, and galaxy formation across cosmic time. In response to the Astro2020 call for science white papers, we provide a high-level synopsis of several major questions in near-field cosmology, the potential for transformative progress in the next decade and beyond, and mechanisms for achieving these goals. Our recommendations include: (i) obtaining kinematic information from large samples of individual line-of-sight and transverse velocities of stars and from spatially resolved gas observations in nearby dwarf galaxies to directly inform our understanding of dark matter interactions in the deeply nonlinear regime; and (ii) an intensive program of optical imaging and UV and optical spectroscopy of resolved stars in low-mass galaxies, including low-metallicity massive stars and HII regions (which are central to understanding faint galaxies at all earlier epochs). Realizing these aims requires a large aperture ($\gtrsim 9$m) UV/optical space telescope, a two-fold community investment in ground-based optical spectroscopy through both ELTs and a dedicated wide-field spectroscopic survey on a 10m class telescope, and radio facilities that can measure high-precision, spatially resolved rotation curves of low-mass galaxies. Finally, we suggest that a servicing mission to extend the life of HST beyond 2025 should be seriously explored.