The main goal of this paper is to use ultraviolet photometry from the Galaxy Evolution Explorer (GALEX) satellite to study chromospheric activity among red giant and core-helium-burning (CHeB) stars. Correlations are sought between several chromospheric and coronal activity indicators for giants that were detected in soft X-rays by the ROentgen SATellite. There is an evident correlation between
log
(
L
X
/
L
bol
)
and
log
(
F
Mg
II
/
F
bol
)
, where L
X is the X-ray luminosity and F
Mg II
is the flux from Mg ii
h and k emission lines, although there is substantial scatter. Using GALEX far-ultraviolet (FUV) magnitudes, the relationship between an FUV-excess parameter (a proxy for chromospheric and transition-region emission lines) and X-ray luminosity is documented. Correlations found herein are not strong, and may be linked to binarity. There is a varied range of FUV emission among CHeB stars, indicating that giants in this phase may be exhibiting differences or cycles in outer atmospheric activity. Additionally, efforts were made to constrain a relationship between excess FUV and near-ultraviolet emission and the projected surface rotation velocity,
v
sin
i
, of giants. No clear relationship was found. Obscurities in this relationship may result from a spread in
sin
i
inclinations, and/or too few stars with
v
sin
i
>
7
km s−1.