Martial Arts, Taekwondo

it is very simple;yet dangerous to develope vey much speed quickly.
to develope speed,simply strike as though your arms/legs were whips.another method in kung fu utilizes a circular striking technique.
i am a 9th degree red belt;and,it took years to develope the art of high speed striking.
soon,i hope to have a video at youtube for multiple technique training.
the main thing to remember is:striking at high speeds places considerable strain on muscles and bones.to strike too quickly may result in broken bones.
do not extend your arms/legs all the way.that may result in broken bones quickly.
while snapping the arm or leg similarly to a whip,move in a small circle to centrifugate the movement.relax until striking;then tense up at the strike impact;then,relax again while retracting your arm/leg.
this technique will quickly develope your speed.

I presume you refer to striking speed, as opposed to reaction speed.
It's not complicated to develop speed, but it requires work.

The first place I would look, not knowing you, is in general fitness. Fat does not move itself, so the less fat the better. Stronger muscles are faster muscles, so strength training is a must. I also happen to like plyometrics.

AFTER you have a level of fitness and strength, the best tools for developing a specific technique would be 1) a solid wall and 2) a striking target.

Let's say you want a faster side kick.

First, practice the first part of the side kick (the "chamber"), drawing up your knee as quickly as possible. Do this for 1 or 2 minute rounds. Then practice the 3rd part of the kick (the "re-chamber" from an extended position). Use the wall for balance if you need to. This is frequently overlooked. Practice timed rounds as before.

To practice the 2nd part of the kick - the extension of the leg, you willneed a target - maybe a bag. You should never practice for speed in the air. While it can result in hyper extension, I find the real trouble much more subtle and insidious. When kicking or punching air, you must use your antagonistic muscles to stop the technique to avoid injury. This trains you to actually fire muscles that SLOW your technique with each strike.

Anyway, from a chambered position, just kick the bag as fast as you can, with no regard for power. Focusing on hitting "hard" generally results in tension, which slows you down.

You can also practice isometrics on the wall by stabilizing yourself with your kicking foot against the wall. Push isometrically from a fully chambered, 1/2 extended, and 3/4 extended leg. Do about 30 seconds per position. This increases your strength in each position. Stronger muscles are faster.

You can do this with most TKD techniques. I have found these ideas most useful.