(Abridged) We present the first results from our spectroscopic survey of the environments of strong gravitational lenses. The lens galaxy belongs to a poor group of galaxies in six of the eight systems in our sample. We discover three new groups associated with the lens galaxies of BRI 0952-0115 (five members), MG 1654+1346 (seven members), and B2114+022 (five members). We more than double the number of members for another three previously known groups around the lenses MG 0751+2716 (13 total members), PG 1115+080 (13 total members), and B1422+231 (16 total members). We determine the kinematics of the six groups, including their mean velocities, velocity dispersions, and projected spatial centroids. The velocity dispersions of the groups range from 110 +170, -80 to 470 +100, -90 km/s. In at least three of the lenses -- MG0751, PG1115, and B1422 -- the group environment significantly affects the lens potential. These lenses happen to be the quadruply-imaged ones in our sample, which suggests a connection between image configuration and environment. The lens galaxy is the brightest member in fewer than half of the groups. Our survey also allows us to assess for the first time whether mass structures along the line of sight are important for lensing. We first show that, in principle, the lens potential may be affected by line-of-sight structures over a wide range of spatial and redshift offsets from the lens. We then quantify real line-of-sight effects using our survey and find that at least four of the eight lens fields have substantial interloping structures close in projection to the lens, and at least one of those structures (in the field of MG0751) significantly affects the lens potential.
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