When I move my object off the preview panel, the bounding box and handles disappear.

Your screenshot shows you moving the layer outside of the Composition Panel. Use Ctrl/Cmnd plus the minus key to scale down the Comp frame, or press and hold the Space Bar to bring up the hand tool, or just press "h" and move the comp frame, or set the Magnification ratio in the bottom left corner of the comp panel to a smaller value so you can see outside the comp frame.

These are all basic UI workflows. Learn more by accessing the Window/Workspace/Learn workspace, by clicking the Learn area on the After Effects Splash screen, or spending some time with the User Guide