When and how drastically would gravity shift for the riders of a space elevator?

OK, lets say I was a passenger on a space elevator, I start from the surface of Earth and climb my way up the tether (or ribbon), eventually I will hit geostationary orbit (the center of mass for the elevator system), when that happens, because of centrifugal force, would gravity simply begin to slow shift so up becomes down? Or would it only happen at the far end where the counterweight/space station would be? If and when it did happen, how strong would that gravity be? Any physicists out there?