We present spatially resolved echelle spectroscopy, obtained at high spectral resolution, for 15 multiple-shell planetary nebulae. Most exhibit faint detached halos (IC 1295, MA 3, M 2-2, M 2-40, NGC 6804, NGC 6826, NGC 6884, NGC 6891, NGC 7662, PM 1-295, and Vy 2-3). Furthermore, we have included some with attached shells (IC 1454, K 1-20, K 3-73, and PM 1-276) to allow comparison of the kinematic properties of the two subclasses of multiple-shell planetary nebulae. In addition, some of the nebulae in our sample show a triple-shell structure, composed of the bright main nebula and a combination of two attached shells (PB 9), one attached shell and one detached halo (NGC 6826, NGC 6891, NGC 7662, and Vy 2-3), or two detached shells (NGC 6804). A new method for computing the expansion velocities of those shells that do not show line splitting has been developed. This method assumes a thick-shell model and uses the observed Hα emission brightness profile to compute the volume emissivity dependence, ɛ(r), with the distance from the center of the nebula. The expansion velocity is then worked out by modeling how much the width of a the Hα line decreases with the radius of the shell. The radial velocity, expansion velocities of each shell, and turbulence contribution to the line width are presented. The expansion velocity of the detached halos spans from 12 to 30 km s^-1. It is worth noting that the expansion velocities obtained by this method are greater than if they were computed with a thin-shell model, as has previously been done. In relation to the attached shells, their expansion velocities span from 10 to 30 km s^-1. When the expansion velocities of the outer attached shells are related to the ellipticity of the inner shells, a trend toward faster expansion of the outer than the inner shells at higher ellipticities is found. The turbulent contribution to the line width has also been established. It is smaller for halos (0 km s^-1 <= sigma_tur <= 6 km s^-1) than for attached shells (0 km s^-1 <= sigma_tur <= 15 km s^-1). This suggests that large-scale hydrodynamic processes are more important in attached shells than in detached halos. We have also studied the kinematics of the detached halos whose morphology is perturbed from a round shape to a dipole asymmetry, indicating its interaction with the surrounding interstellar medium. We found systematic differences between the kinematical behavior of the enhanced edge of the halo and the opposite side in these cases, thus revealing the kinematic effect of the interaction of the halos with the interstellar medium.