Given a set C of set P of sets R, find the smallest set S such that at least one subset R from each P is a subset of S
Your problem is NP-hard, by reduction from Monotone SAT.

Recall that Monotone SAT is (the decision version of) the following problem: given a CNF formula $\varphi$, find a satisfying assignment with the fewest variables set to true.

Now, map each clause, say $x_{i_1} \lor x_{i_2} \lor \dots \lor x_{i_k}$, in $\varphi$ to a set $P=\{R_1,R_2,\dots,R_k\}$ where $R_1=\{x_{i_1}\}$, $R_2=\{x_{i_2}\}$, etc.  Finally, form $C$ to be the set of $P$'s obtained in this way from the clauses of $\varphi$.  Then a set $S$ satisfies $C$ iff it corresponds to a satisfying assignment for $\varphi$ (including the element $x_i \in S$ corresponds to setting $x_i$ to true in the assignment), so the minimal-size set $S$ that satisfies C corresponds to the minimal-size satisfying assignment for $\varphi$.

Since Monotone SAT is NP-hard, your problem is NP-hard, too.

If you need to try to find an exact solution in practice, you could try formulating your problem as an instance of SAT and feed it to a SAT solver (using binary search over the size of $S$), or you could formulate it as an instance of integer linear programming and feed it to an ILP solver.