What is the notation for saying $S$ is a subring of $R$? I can't find any. For saying $H$ is a subgroup of $G$ we have notation but it seems none exists for subrings.

If $S$ is a subring of $R$ we write $S\hookrightarrow R$. The hooked arrow means 'monomorphism', which encapsulates the idea of $S$ being isomorphic to a sub-object of $R$ which is in this case a subring.
EDIT
After the discussion in quid's answer I will explain this a little. 
The symbol '$\subset$' means 'subset'. The symbol '$\hookrightarrow$' means 'subobject'. Hence the latter can be used to denote a subgroup, a subring, a sub(topological/metric/&c.)space and indeed even a subset. 
