We study four lines of sight that probe the transition from diffuse molecular gas to molecular cloud material in Taurus. Measurements of atomic and molecular absorption are used to infer the distribution of species and the physical conditions toward stars behind the Taurus Molecular Cloud (TMC). New high-resolution spectra at visible and near-IR wavelengths of interstellar Ca ii, Ca i, K i, CH, CH+, C-2, CN, and CO toward HD 28975 and HD 29647 are combined with data at visible wavelengths and published CO results from ultraviolet measurements for HD 27778 and HD 30122. Gas densities and temperatures are inferred from C-2, CN, and CO excitation and CN chemistry. Our results for HD 29647 are noteworthy because the CO column density is 10(18) cm(-2) while C-2 and CO excitation reveals a temperature of 10 K and a density of similar to 1000 cm(-3), more like conditions found in dark molecular clouds. Similar results arise from our chemical analysis for CN through reactions involving observations of CH, C-2, and NH. Enhanced potassium depletion and a reduced CH/H-2 column density ratio also suggest the presence of a dark cloud. The directions toward HD 27778 and HD 30122 probe molecule-rich diffuse clouds, which can be considered CO-dark gas, while the sight line toward HD 28975 represents an intermediate case. Maps of dust temperature help refine the description of the material along the four sight lines and provide an estimate of the distance between HD 29647 and a clump in the TMC. An appendix provides results for the direction toward HD 26571; this star also probes diffuse molecular gas.