Many players move their playing hand and arm first when moving to a ball, when the first thing that should move are the feet. If you move your arm first, several very bad things are happening. First, it means you are slower in moving into position since you haven't started moving your feet first. Second, you will tend to reach for the ball--since by moving your hands first you are reaching for the ball--and so will generally hit an awkward shot. And third, by reaching for the ball, you go off balance, with your weight on the foot in the direction you are reaching, and so will have great difficulty moving the feet at that point, since you are now weighted down on the lead foot, the very one that should be moving first in the direction you need to move. (Try this and you'll see what I mean.)
Instead, focus on staying balanced. This means your weight should always be between your feet when moving. There might be a weight shift once you have moved into position (especially on forehand shots), but not before. Balance allows you to move quickly in either direction and to glide into position. It makes that first step very easy. (Remember to step first with the right to move right, with the left to move left.)
Most coaches do tell you to move your feet first, not the playing hand and arm, but players often have trouble following this. I've found that if you coach the player to focus on the balance aspect, then they more naturally move the feet first, since balance is what allows this to happen easily, and it's the playing hand and arm movement that takes a player off balance--so by focusing on balance, they stop doing that. Give it a try, and get your game into balance!