How do I move rests?

Entering that doesn't involve moving rests at all. And even then, the way to go is not to move the silence (rest), but to move the sound (notes). A rest is only what remains if no notes are in that spot.

You start by entering the half rest, then the quarter rest, then the notes. Entering a rest is done exactly the same as entering a note (see also https://musescore.org/en/handbook/note-input). You first select the desired duration and then input (mouse/keyboard) the element.