Since you have been given a distribution for S^2 in terms of chi-square, you want to find the expectation of S by itself.
Recall that the expectation of a function of a random variable E[g(X)] for a continuous distribution is Integral (-infinity,infinity) g(x)f(x)dx. Now X = S^2. Let g(X) = SQRT(X) and now given the expectation theorem, you can find out what E[SQRT(S^2)] = E[S] is.
Also be aware that n and sigma^2 are constants and not random variables so E[S^2] = sigma^2/(n-1)*E[chi-square_n_distribution].
Also remember that the expectation of a known distribution always returns some number whether its an actual number or just an expression in terms of non-random constants.