I second what Kacey said, and to that add this:
(1) Pick up your feet with every step. Unless you're doing bassai so, you don't need to be dragging your feet. Bring your stepping knee up fast, and you'll be moving fast.
(2) On hand techniques, remember that the hand going out can only go as fast as the hand coming back to your ribs. As one hand goes out, the other has to come in as fast. The faster the one goes in, the faster the other goes out.
(3) Don't look at your hands. You're eyes should stay straight ahead. If you have to look at your hands, you're not paying as much attention to the task at hand. You need to react quickly.
(4) Never punch from the hips. Punch from the ribs. That little time it takes to raise your hand to center level from your hips and punch it out can make a big difference. Your hand has to raise anyway, whether it does so before the punch or during the punch. Keep it on your ribs, and it's already raised before you even need to punch. If you're fighting, keep your back hand near your solar plexus, for the same reason. Dropping your hands is never a good idea (unless it's a low block, in which case block and then counter!)
Hope that's helpful.
Tang Soo!