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 2800 A 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 at K2 III to ~20 km/s at M5 III. We find two new detections of astrospheric absorption, for Sigma Pup (K5 III) and Gamma Eri (M1 III). This absorption signature had previously only been detected for Alpha 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 Gamma Eri astrosphere is provided to explore this further.