We utilize the Sloan Digital Sky Survey Baryon Oscillation Spectroscopic Survey (SDSS-BOSS) galaxies and its overlap with approximately 416 sq degrees of deep grizy-band imaging from the Subaru Hyper Suprime-Cam Survey (HSC). We perform measurements of three two-point correlations which form the basis of the cosmological inference presented in our companion papers, Miyatake et al. and Sugiyama et al. We use three approximately volume limited subsamples of spectroscopic galaxies by their -band magnitude from the SDSS-BOSS: LOWZ (), CMASS1 () and CMASS2 (), respectively. We present high signal-to-noise ratio measurements of the projected correlation functions of these galaxies, which is expected to be proportional to the projected matter correlation function on large scales with a proportionality constant dependent on the bias of galaxies. In order to help break the degeneracy between the amplitude of the matter correlation and the bias of these spectroscopic galaxies, we use the distortions of the shapes of fainter galaxies in HSC due to weak gravitational lensing, to measure the galaxy-galaxy lensing signal, which probes the projected galaxy-matter cross-correlation function of the SDSS-BOSS galaxies. We also measure the cosmic shear correlation functions from HSC galaxies which is related to the projected matter correlation function. We demonstrate the robustness of our measurements by subjecting each of them to a variety of systematic tests. Our use of a single sample of HSC source galaxies is crucial to calibrate any residual systematic biases in the inferred redshifts of our galaxies. We also describe the construction of a suite of mocks: (i) spectroscopic galaxy catalogs which obey the clustering and abundance of each of the three SDSS-BOSS subsamples, and (ii) galaxy shape catalogs which obey the footprint of the HSC survey and have been appropriately sheared by the large-scale structure expected in a Cold Dark Matter model. We use these mock catalogs to compute the covariance of each of our observables.