We report the first ensemble results from the Far Ultraviolet Spectroscopic Explorer survey of molecular hydrogen in lines of sight with A_V>~1 mag. We have developed techniques for fitting computed profiles to the low-J lines of H₂, and thus determining column densities for J=0 and J=1, which contain >~99% of the total H₂. From these column densities and ancillary data we have derived the total H₂ column densities, hydrogen molecular fractions, and kinetic temperatures for 23 lines of sight. This is the first significant sample of molecular hydrogen column densities of ~10²¹ cm^-2, measured through UV absorption bands. We have also compiled a set of extinction data for these lines of sight, which sample a wide range of environments. We have searched for correlations of our H₂-related quantities with previously published column densities of other molecules and extinction parameters. We find strong correlations between H₂ and molecules such as CH, CN, and CO, in general agreement with predictions of chemical models. We also find the expected correlations between hydrogen molecular fraction and various density indicators such as kinetic temperature, CN abundance, the steepness of the far-UV extinction rise, and the width of the 2175 Å bump. Despite the relatively large molecular fractions, we do not see the values greater than 0.8 expected in translucent clouds. With the exception of a few lines of sight, we see little evidence for the presence of individual translucent clouds in our sample. We conclude that most of the lines of sight are actually composed of two or more diffuse clouds similar to those found toward targets like ζ Oph. We suggest a modification in terminology to distinguish between a ``translucent line of sight'' and a ``translucent cloud.''