In an edge-colored graph, a traversal cost occurs at a vertex along a path when consecutive edges with different colors are traversed. The value of the traversal cost depends only on the colors of the traversed edges. This concept leads to two global cost measures, namely the \emph{reload cost} and the \emph{changeover cost}, that have been studied in the literature and have various applications in telecommunications, transportation networks, and energy distribution networks. Previous work focused on problems with an edge-colored graph being part of the input. In this paper, we formulate and focus on two pairs of problems that aim to find an edge coloring of a graph so as to minimize the reload and changeover costs. The first pair of problems aims to find a proper edge coloring so that the reload/changeover cost of a set of paths is minimized. The second pair of problems aim to find a proper edge coloring and a spanning tree so that the reload/changeover cost is minimized. We present several hardness results as well as polynomial-time solvable special cases.