Constacyclic codes of length $p^s$ over $R=mathbb{F}_{p^m}+umathbb{F}_{p^m}+u^2mathbb{F}_{p^m}$ are precisely the ideals of the ring $frac{R[x]}{}$. In this paper, we investigate constacyclic codes of length $p^s$ over R. The units of the ring R are of the forms ${gamma}$, ${alpha}+u{eta}$, ${alpha}+u{eta}+u^2{gamma}$ and ${alpha}+u^2{gamma}$, where ${alpha}$, ${eta}$ and ${gamma}$ are nonzero elements of $mathbb{F}_{p^m}$. We obtain the structures and Hamming distances of all (${alpha}+u{eta}$)-constacyclic codes and (${alpha}+u{eta}+u^2{gamma}$)-constacyclic codes of length $p^s$ over R. Furthermore, we classify all cyclic codes of length $p^s$ over R, and by using the ring isomorphism we characterize ${gamma}$-constacyclic codes of length $p^s$ over R.