How do I move the camera in any perspective mode so the reference image doesn't vanish? I use the Shift + F method to move my camera when I'm modeling(which makes my background image/reference vanish) but lately I watched this person who could move the camera without the BG image vanishing:
https://www.youtube.com/watch?v=iI3zaenro-M

This would be incredibly helpful but I simply don't know how to do that and my google searches have turned up nothing, so I really need some help from the community on this :)
Your background image will vanish as soon as you leave orthographic view or rotate the camera to an angle the background image isn´t set for.

Just stay in orthographic view (Numpad 5) and use Front View (Numpad 1), Side View (Numpad 3) and Top View (Numpad 7) to change the directions; for the opposite of those views press Ctrl as well.

To move your current view without rotating just press Shift and drag your view with Middle Mouse Button