What to do on rest period (push-up)

The rest period is the time that you should be recovering, to get ready for the next set. But you can skip the rest period (by tapping it) if you wish, and jump into the next set whenever you want.

Having said that, if you want to use that rest period to get in an even more grueling workout, you can definitely stay in the pushup position (plank) if you want! Completely up to you.