Hi Slicepong,
I’d say the body is a key area to effective movement but also being low and active with the legs is so important. You neeed to be low, on the balls of your feet and ready to make those quick adjustments. Bouncing on your toes like you said is very good and makes you ready to move,you should also try to focus on really engaging your quad muscles which are the biggest muscles in the legs and give you the power to move quickly.
Lastly when returning serve you should have a look at your distance from the table to see if you are getting rushed by being too close. Also trying to stay relaxed and not tighten up in the upper body can really help improve your movement.
