A very important, yet overlooked aspect of moving on the tennis court has to do with when to get your feet going and when to be stopped.
Whenever you, your partner or your opponent are about to hit the ball (the forward swing is starting), your feet should be set.
Using the split step just before contact is a great way to do this. Steffi Graf was one of the best at using the split step to regain her balance just before the hit of the ball, and this enabled her to change direction and move very efficiently to the opponent's shot.
Whenever the ball is in the air between the point of contact on each side of the net, all players should be moving. The movement can be forward, back, towards the direction of the shot, or to the middle of the court. If you time your movement as described here, you will be able to react quickly to the ball and and also recover very quickly after the shot is hit.
This will result in better coverage of the court overall.