We present several different measurements of the velocities of structures within the circumstellar envelope of SN 1987A, including the inner, equatorial ring (ER), the outer rings (ORs), and the diffuse nebulosity at radii 5 pc, based on CTIO 4 m and Hubble Space Telescope (HST) data. A comparison of STIS and WFPC2 [N II] λ6583 loci for the rings show that the ER is expanding in radius at 10.5 ± 0.3 km s^-1 (with the northern OR expanding along the line of sight at ~26 km s^-1, and for the southern OR, ~23 km s^-1). The best fit to CTIO 4 m echelle spectra of the [N II] λ6583 line show the ORs expanding at ~23 km s^-1 along the line of sight. Accounting for inclination, the best fit to all data for the expansion in radius of both ORs is 26 km s^-1. The ratio of the ER to the OR velocity is nearly equal to the ratio of the ER to the OR radius, so the rings are roughly homologous, all having been created ~20,000 yr before the supernova (SN) explosion. This makes the previously reported, large compositional differences between the ER and ORs difficult to understand. Additionally, a grid of long-slit 4 m echelle spectra centered on the SN shows two velocity components over a region roughly coextensive with the outer circumstellar envelope extending ~5 pc (20'') from the SN. One component is blueshifted ~10 km s^-1 relative to the systemic velocity of the SN, while the other is redshifted by a similar amount. These features may represent a bipolar flow expanding from the SN, in which the ORs are propelled 10-15 km s^-1 faster than that of the surrounding envelope into which they propagate. The kinetic timescale for the entire nebula is 350,000 yr (and probably more, since material may be accumulating in an outer contact discontinuity). The kinematics of these different structures constrain possible models for the evolution of the progenitor and its formation of a mass-loss nebula.