We present the target selection, observations, and data reduction and analysis process for a program aimed at discovering numerous, weak (equivalent width ≤ 100 mÅ) Lyα absorption lines in the local Universe (0.003 ≤ z ≤ 0.069). The purpose of this program is to study the physical conditions of the local intergalactic medium, including absorber distributions in Doppler width and H I column density, redshift evolution of absorber number density, line-of-sight two-point correlation function, and the baryonic content and metallicity. By making use of large-angle, nearby galaxy redshift surveys, we will investigate the relationship (if any) between these Lyα absorbers and galaxies, superclusters and voids. In Paper I, we present high resolution (∼ 19 km s −1 ) spectroscopic observations of 15 very bright (V ≤ 14.5) AGN targets made with the Goddard High Resolution Spectrograph (GHRS) aboard the Hubble Space Telescope (HST). We find 81 definite (≥ 4σ) and 30 possible (3 − 4σ) Lyα absorption lines in these spectra, which probe a total pathlength of 116,000 km s −1 (∆z ∼ 0.4) at very low redshift (z ≤ 0.069) and column density (12.5 ≤ log N HI ≤ 14.5). We found numerous metal lines arising in the Milky Way halo, including absorption from high velocity clouds in 10 of 15 sightlines and numerous absorptions intrinsic to the target AGN. Here, we describe the details of the target selection, HST observations, and spectral reduction and analysis. We present reduced spectra, absorption line lists, "pie diagrams" showing the known galaxy distributions in the direction of each target, and nearest galaxy tables for each absorber. In Papers II and III, we use the data presented here to determine the basic physical characteristics of the low-z Lyα forest and to investigate the relationship of the absorbers to the local galaxy distribution.