Using the new Safari I can move between tabs by swiping left and right with two fingers. When I try to do the same using Chrome it takes me back and forward on the same page I'm at.
How can I change the gesture for Chrome?
Try BetterTouchTool. It allows you to have much more granular control over gestures than the standard OS X trackpad preferences, and you can do things like set per-application configurations for gestures. Even if there's no back/forward action for Chrome specifically, you should be able to set the gestures to activate the Back/Forward menu items or shortcuts.