In this context, I think this sounds more fluent / natural: "The best thing you can do is keep on moving." So yes it means to not stop doing things. Like 'don't give up!' or 'don't stop trying'.

Hi Emarbe, 'Keep on the move' really means that you need to physically keep moving. So it wouldn't be appropriate in this context. Maybe you can say: The best thing you can do is (to) keep on going. or The best thing you can do is (to) keep on trying. Best wishes, Neil