In positive characteristic, quotient singularities need not be rational. For an example, see Artin's paper "Wildly ramified Z/2-actions in dimension two".
In characteristic zero, quotient singularities are log terminal, and log terminal singularities are rational.
The first fact is a simple exercise on discrepancies (which I think first appeared in a paper of Shokurov), basically a straightforward application of Riemann-Hurwitz ramification formula.
For the second one, an elementary proof is contained in the book of Koll'ar and Mori.