The distinguishing chromatic number of a graph \( G \) is the least integer, \( \chi_D(G) \), for which \( G \) has a coloring of its vertices so that adjacent vertices receive different colors, and the identity is the only automorphism of \( G \) that preserves vertex colors. Our focus is on determining the distinguishing chromatic numbers of wreath products of graphs, extending the work of Tang. We prove that if \( C_n \) is a cycle with \( n \) vertices and \( P_n \) is a path with \( n \) vertices, then \( \chi_D(C_n[G]) \) and \( \chi_D(P_n[G]) \) can be found for any connected graph \( G \). We also obtain an upper bound on \( \chi_D(T[G]) \) when \( T \) is a tree and \( G \) is any connected graph. Some of our results depend on the notion of inequivalent colorings. Cheng introduces inequivalent colorings and provides a formula for computing the number of inequivalent distinguishing \( k \)-colorings of a rooted tree. We add to this work by obtaining an expression for computing the number of inequivalent distinguishing \( k \)-colorings of a cycle.