In the stochastic processes literature, the idea of the mean hitting time describes the average amount of time a process takes to move from one state to another state (or set of states). However, due to the distributions of hitting times having very high variance, it is clear that the mean is often not a good measure. For this reason, we investigate the distributions of these hitting times. For purposes of simplicity and tractability, we considered random walks on finite graphs that were the same from the perspective of every vertex (or vertex transitive). Through working with the spectral properties of these graphs, we were able to arrive at several formulas to express the hitting time distribution between any two states on individual simple graphs (like cycles, paths, complete bipartite, and complete). As a final result, we extended our analysis to derive an infinite series where the coefficients corresponded to the hitting time distribution.