Second, it depends on what you mean by improving your movement speed - but the first thing I would suggest is your breathing - the way to strike or kick faster is to breath faster.
(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!)
Same thing they teach you when you're first learning a musical instrument:
Get the basics down while doing it slowly. When you've "perfected" your technique, speed will develop naturally. It's more important to do kicks and punches correctly as opposed to merely flailing your arms and legs wildly at an opponent.
Its called practice ;-) The better you know the movements, and transitions, the more they become "grafted to the body" so you may move with instinctual speed, and accuracy...
If your technique is there and your just don't feel your fast enough you can try practicing your punches while holding weights and kicking with ankle weights on, also try resistance bands we have done a drill in class were two people of the same size are paired up each with one end of a resistance band (bungee cord with a velco strap on each end) attached to their ankle you step apart until there is a little tension on the band then one person kicks while the other person stands still then they switch. The person kicking has to work harder to execute the kick and it forces proper technique as you have no choice but to load, at the same time the person who is just standing has to work to keep their balance. It works for front side and round kicks but you need to be carefull how you stand for side and round so you don't get tangled in the cord. Hope this helps.
The most common problem I see is that students are tense, relax in your movement, relax your strikes until just before impact. Most beginners are slow because they are working against themselves.