In general, the way to learn to get fast is to go slow.
Seriously.
Being fast comes from being smooth, and being smooth starts with being slow. So for any physical skill, be it drawing your gun from a holster, shooting quick shot strings, moving with a loaded gun, etc., start off by doing it slowly. And while your doing it think about and concentrate on each component of the task, each step of the process, and focus on doing each step perfectly. Keep repeating the process and keep focusing on each step and doing it perfect. What you are doing is programing you body to ultimately perform the task reflexively, without conscious thought.
As you keep repeating the task, focusing on each part of the process, the "programing" keeps getting better, your actions get smoother; and as you get smoother, you also get faster. So slow is smooth, and smooth is fast.
To illustrate this, let's say you want to learn to draw your gun from the holster. The more-or-less standard presentation from a strong side belt holster, as taught most places these days, goes roughly like this:
[1] You want to achieve a full firing grip before withdrawing the pistol from the holster. You should not have to shift your grip. Throughout the draw stroke, until you are actually going to fire the gun, the trigger finger stays off the trigger, outside the trigger guard and indexed along the frame.
[2] While the strong hand is moving to grip the pistol, the weak hand is placed flat on the abdomen near the same level as the grip of the pistol. This helps assure that the weak hand isn't swept by the muzzle and also puts the weak hand in position to take grip the pistol over the strong hand.
[3] The pistol is withdrawn straight upwards from the holster, and the muzzle is rotated toward the target after it clears the holster. If using 1911, Browning High Power, or some other gun with a safety engaged, the safety may be disengaged here, but the trigger finger remains off the trigger, outside the trigger guard and indexed along the frame.
[4] When the muzzle is rotated toward the target the strong hand is at about the level of the strong side pectoral muscle and the strong hand is held against the side with the muzzle pointed to the threat. If the threat is very close, 1-2 yards, the gun may be fired from this position. This is called the retention position.
[5] At the retention position, the weak hand comes up to assume its part of the grip. The two hands then together extend the gun either fully up to shooting position or partially at a downward angle to the low ready position, depending on the circumstances.
[6] The gun is holstered by following those steps in reverse. I have been taught to follow these steps whenever removing my gun from, or placing my gun in, the holster.
[7] I've also been taught to begin moving my strong hand to the gun from about my belt buckle. The thing is that if I'm carrying my gun concealed I will need to displace my covering garment to gain access to the gun. If I sweep my strong from approximately mid line I automatically sweep aside my covering garment.
Begin, with a weapon you have confirmed to be unloaded, and perform the draw slowly, concentrating on each of these steps. Then do it again and again and again, over and over. You can graduate to doing this at the range, if the range will permit the use of a holster, and add live fire. So you'd draw, still slowly concentrating on doing each strep perfectly, and firing the gun when on target. The bad news is that it takes about 5,000 repetitions to make an action reflexive. The good news is that dry practice count, as long as you're doing everything perfectly. It's perfect practice that makes perfect.
Two key words here: smooth and control. The goal is to do this smoothly. If one concentrates on being smooth and practice over and over again, he will get fast. Speed comes from smoothness and no wasted motion. And one must be in control at all times. At lot is going on, and a misstep on the presentation can be devastating. But by being smooth you retain control, and by being smooth you become fast. And by being smooth and in control you will be accurate.
Finally, some professional training is always a good idea. There is really no good substitute for a qualified instructor watching you and helping you to correct things.