If I have an artboard with an image covering the whole background, how can I move it around? There is nowhere to grab the board and cmd click to move it as usual. If I try to move it, I just end up dragging the image off the artboard.
I suggest locking the image and move the artboard as you need. Select the image then go to the menu as show in the pic. To unlock afterwards just to to the same menu for unlock and all locked objects will be released.
Keyboard shortcut to lock is Cmd+2 (for mac) or Ctrl+2 (for pc).
To release is Cmd+Alt+2 (for mac) or Ctrl+Alt+2 (for pc).
You can use the artboard title as a handle. Click the artboard title, then you can drag it wherever you'd like.