We report on an ultraviolet spectroscopic survey of red giants observed by the Hubble Space Telescope, focusing on spectra of the Mg II h & k lines near 2800Å in order to study stellar chromospheric emission, winds, and astrospheric absorption. We focus on spectral types between K2 III and M5 III, a spectral type range with stars that are noncoronal, but possessing strong, chromospheric winds. We find a very tight relation between Mg II surface flux and photospheric temperature, supporting the notion that all K2-M5 III stars are emitting at a basal flux level. Wind velocities (V w ) are generally found to decrease with spectral type, with V w decreasing from ∼ 40 km s −1 at K2 III to ∼ 20 km s −1 at M5 III. We find two new detections of astrospheric absorption, for σ Pup (K5 III) and γ Eri (M1 III). This absorption signature had previously only been detected for α Tau (K5 III). For the three astrospheric detections the temperature of the wind after the termination shock correlates with V w , but is lower than predicted by the Rankine-Hugoniot shock jump conditions, consistent with the idea that red giant termination shocks are radiative shocks rather than simple hydrodynamic shocks. A full hydrodynamic simulation of the γ Eri astrosphere is provided to explore this further.