The Ly{alpha} forest has opened a new redshift regime for cosmological investigation. At z>2 it provides a unique probe of cosmic geometry and an independent constraint on dark energy that is not subject to standard candle or ruler assumptions. In Paper I of this series on using the Ly{alpha} forest observed in pairs of QSOs for a new application of the Alcock-Paczynski test, we present and discuss the results of a campaign to obtain moderate-resolution spectroscopy (FWHM~2.5{AA}) of the Ly{alpha} forest in pairs of QSOs with small redshift differences ({Delta}z<0.25, z>2.2) and arcminute separations ({Theta}<5'). This data set, composed of seven individual QSOs, 35 pairs, and one triplet, is also well suited for future investigations of the coherence of Ly{alpha} absorbers on ~1Mpc transverse scales and the transverse proximity effect. We note seven revisions for previously published QSO identifications and/or redshifts.