My Son has been working on this for a while. One easy place to start is by doing 10 yard sprints. You will improve your time most significantly by focusing first on your start, getting to full speed in 3 steps.
Working on you legs and core (weight training) is another place to spend some quality time.
A drill to increase hand quickness that we have done you can begin without a glove on. Have someone throw to you, not hard, you don't have a glove on...anyway, instead of catching the ball with your glove hand, deflect it off your palm into your throwing hand. Start with your hands close together. As you improve you will be able to deflect the ball 3' into your throwing hand with no problem, without even looking you will know where the ball will be going.
Once you have mastered this bare hands, put your glove on and develop your skill deflecting the ball from the glove into your throwing hand. You will need a glove that is relatively "flat" not a deep pocket. Eventually you will find that you no longer "catch" a grounder when fielding (you never close your glove) but instead you will deflect the ball into your throwing hand. This will have a dramatic impact on the speed of getting the ball in and out of your glove. You must also play very low to the ground so that your center of gravity remains low.
Obviously there will be some plays such as diving for a line drive in the gap where you will want (need) to close your glove.