The giant southern stream (GSS) is the most prominent tidal debris feature in M31's stellar halo and covers a significant fraction of its southern quadrant. The GSS is a complex structure composed of a relatively metal-rich, high-surface-brightness "core" and a lower metallicity, lower-surface-brightness "envelope." We present spectroscopy of red giant stars in six fields in the vicinity of M31's GSS (including four new fields and improved spectroscopic reductions for two previously published fields) and one field on stream C, an arc-like feature seen in star-count maps on M31's southeast minor axis at R ~ 60 kpc. These data are part of our ongoing Spectroscopic and Photometric Landscape of Andromeda's Stellar Halo survey of M31 using the DEIMOS instrument on the Keck II 10 m telescope. Several GSS-related findings and measurements are presented here. We present the innermost kinematical detection of the GSS core to date (R = 17 kpc). This field also contains the inner continuation of a second kinematically cold component that was originally seen in a GSS core field at R ~ 21 kpc. The velocity gradients of the GSS and the second component in the combined data set are parallel over a range of ΔR = 7 kpc, suggesting that this may represent a bifurcation in the line-of-sight velocities of GSS stars. We present the first kinematical detection of substructure in the GSS envelope (S quadrant, R ~ 58 kpc). Using kinematically identified samples, we show that the envelope debris has a ~0.7 dex lower mean photometric metallicity and possibly higher intrinsic velocity dispersion than the GSS core. The GSS is also identified in the field of the M31 dwarf spheroidal satellite And I; the GSS in this field has a metallicity distribution identical to that of the GSS core. We confirm the previous finding of two kinematically cold components in stream C, and measure intrinsic velocity dispersions of ~10 and ~4 km s^–1. This compilation of the kinematical (mean velocity, intrinsic velocity dispersion) and chemical properties of stars in the GSS core and envelope, coupled with published surface-brightness measurements and wide-area star-count maps, will improve constraints on the orbit and internal structure of the dwarf satellite progenitor.