The chemical abundances of very metal-poor stars provide important constraints on the nucleosynthesis of the first generation of stars and early chemical evolution of the Galaxy. We have obtained high-resolution spectra with the Subaru Telescope for candidates of very metal-poor stars selected with a large survey of Galactic stars carried out with LAMOST. In this series of papers, we report on the elemental abundances of about 400 very metal-poor stars and discuss the kinematics of the sample obtained by combining the radial velocities measured in this study and recent astrometry obtained with Gaia. This paper provides an overview of our survey and follow-up program, and reports radial velocities for the whole sample. We identify seven double-lined spectroscopic binaries from our high-resolution spectra, for which radial velocities of the components are reported. We discuss the frequency of such relatively short-period binaries at very low metallicity.