Today's focus was on how to move quickly. To move quickly you must relax - breathing out (while moving) helps you relax and become faster. Tension slows you down. When you hit something, you cannot be relaxed, or there will be now power and you may injure yourself. So we instantaneously tense, only at the very end of the technique, and then relax. This instantaneous tensing and relaxing at the end of the technique is how perform decisive techniques - kime waza. Your whole body tenses, not just your arm.
When stepping forward, use your pivot/supporting leg jiku ashi to pull you forward, keeping it bent and remaining at the same height, pulling your rear foot in. Once the rear foot reaches your front foot you are in a state of compression. From here drive yourself with your compressed leg and make your technique.
We went through Kihon Kata, Heian Shodan and Heian Nidan. You can view videos of these kata to help you learn them. The above principals apply to kumite and kata - speed and power are essential to good kata.