Cylindrically symmetric vacuum spacetimes are of immense interest in theoretical physics due to their connection to cosmic strings hypothesized in quantum field theory. In this article, we explore the properties of such a spacetime and provide the complete, exact solution by quadrature to the timelike and lightlike geodesics in it, using the Hamilton-Jacobi formalism. In addition, we compare several properties of massive particle trajectories in relativistic cylindrically symmetric vacuum spacetimes to their nonrelativistic, Newtonian gravitational counterparts. On top of that, we devise a classification scheme and use it to categorize the orbits of massive particles.