  (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.
