How are lowercase k and uppercase K used in general chemistry courses?
In addition to the original answer I also allowed myself to invite lowercase Greek letter kappa κ to the table as typographically it has a similar-looking appearance and add some more definitions:

Lowercase k


$k$ is one of the Miller indices.
$k$ is a symbol for thermal conductivity.


Uppercase K


$\ce{K}$ is a symbol for potassium.
$\pu{K}$ is a symbol for Kaiser, unit of energy $(\pu{1 K} = \pu{1 cm-1})$.
K is a symbol for aminoacid lysine.
K is used in spectrometry to denote spectral lines.
$K$ is sometimes used as a symbol for decadic absorption coefficient.


Lowercase Greek letter kappa κ


κ is used to denote denticity.
$κ$ is a symbol for [specific] conductivity.
$κ$ is a symbol for molar Napierian absorption coefficient.
$κ$ is a symbol for asymmetry parameter.
$κ$  is sometimes used as a symbol for bulk viscosity.
$κ$ is sometimes used as a symbol for magnetic susceptibility.


Uppercase Greek letter kappa Κ


Κ is a symbol for a kappa number characterizing bleachability.


Both lowercase k and lowercase Greek letter kappa κ


$k$ and $κ$ is a symbol for imaginary refractive index.


Both lowercase k and uppercase K


$k$ and $K$ is in general may refer to a coefficient (German "Koeffizient"); indices are often used to denote which one, e.g. mass transfer coefficient $k_\mathrm{d}$, octanol/water partition coefficient $K_\mathrm{ow}$.