  We consider finitely generated free semigroup actions on a compact metric
space and obtain quantitative information on Poincar\'e recurrence, average
first return time and hitting frequency for the random orbits induced by the
semigroup action. Besides, we relate the recurrence to balls with the rates of
expansion of the semigroup's generators and the topological entropy of the
semigroup action. Finally, we establish a partial variational principle and
prove an ergodic optimization for this kind of dynamical action.
