If I have $a \in A$ and $s \in S$ and different function values $R(a)$ which, for instance, could be $$R(a_1) = \{ (s_1,s_2),(s_2,s_3),\ldots \}$$
What is the definition of the function $R$? I guess it is either $$ R : A \to S \times S $$ or $$ R : A \to 2^{S \times S} $$
Is it also O.K. to say $a \in A$ or would it be more correct to say $a_i \in A$ for all $i \in \mathbb{Z}_{>0}$?
If some arbitrary element $a_1\in A$ gets mapped to some set of the form $\{(s_1,s_2),(s_2,s_3),\ldots\}$ where $s_i\in S$, then that function, which you call $R$, is a function from $A$ to $\mathscr{P}(S\times S)$ the powerset of $S\times S$.
Answering your second question, it definitely would be more correct to say $a_i\in A$. Otherwise I may be told that $a\in A$, but that doesn't tell me $a_0\in A$ or $a_i\in A$ for that matter; I only know $a\in A$.