The goal of a Taekwondo competition is to score points by hitting your opponent with kicks or punches. Power isn't as important as speed when you're trying to score points. Quick strikes are harder to dodge and do more damage. To get faster, you have to practice and hone your technique to make yourself more efficient.
Some martial arts movies show characters using impressive drills like catching a fly with their bare hands to improve speed. You're probably better off practicing Taekwondo techniques that have been honed over nearly a thousand years of practice.
One of the best ways to increase your speed is to simply practice. The more you repeat a movement, the more efficient your neuromuscular system becomes. Some Taekwondo competitors practice two to four days per week, one to three hours per session. When you practice, focus on speed and strike as fast as possible. At the same time, keep your technique as clean as possible.
Proper technique can also help increase your speed. In order to be fast you have to maintain your balance throughout a movement. If you don't, you'll be out of position. Learning proper Taekwondo technique will help you maintain your balance.
If you're a beginner or if you're new to speed training, start off slow. Go through the movements you want to practice at a little less than full speed. You'll be able to maintain your balance and coordination much better if you go slower. Over time you'll increase your speed gradually and your form won't suffer as a result.
Training outside of Taekwondo can carry over into martial arts practice. Lifting weights and practicing plyometric drills improves your muscular strength and speed overall, which you can funnel into Taekwondo movements.
Bodyweight plyometric drills are a good place to start. They make you faster without packing on extra muscle that increases your bodyweight. For the upper body, clap push-ups will do the trick.
Increase the speed of your punching muscles, the shoulders, triceps and chest, with clap push-ups. These are significantly harder than regular push-ups, so you can start with your knees on your ground and work up to a regular clap push-up.
HOW TO DO IT: Get into a push-up position. Lower yourself to the ground, then explode up as fast as you can and briefly leave the ground. While you're airborne, clap your hands together, then land on your hands with bent elbows and repeat.
Lateral Skater Jumps
In Taekwondo, you need to be comfortable generating power off of one leg for kicks. In order to build up power and explode off of one leg, you have to have incredible balance and coordination as well as speed. Lateral skater jumps help you build up all of those qualities.
HOW TO DO IT: Start standing in an open space with at least 10 feet to your side. Load up on your left leg and jump to the right. Don't turn your body to either side. Rather, face forward the entire time. Land on your right leg with your knee bent. As you land, sink down onto your right leg to absorb the impact. Then, jump to the left by pushing off your right leg, landing on the left. Repeat for 20 seconds, jumping as high and as far as possible.