I think it's important to be a little more systematic than just try to shoot stuff as fast as you can. Pro football players don't just go out and play football. They break down the skills needed to play their position and drill those skills in isolation, and then string together multiple skills before ever trying to do it in a simulated play. In shooting, it's probably a good idea for newer shooters to spend some time just working on trigger pull while looking at the sights, without any targets to distract you (white wall fundamentals). Single pulls, (in SA and DA if you shoot that type of gun), double pulls, and bill drills, all with the same intensity of grip you use while shooting.
So imho, ONE of the things you should tell a new shooter is 'align the sights, and pull the trigger as fast as you can without disturbing the sights'. For sure, at the same time, the new shooter needs to develop the skill to shoot tight groups, and hit what he is aiming at, but most people should probably be working on both.
Mr. Moto, I agree with you 100%. Breaking down any endeavor into the basic skills is how all things are taught and learned. You start with the most basic skills and master those before attempting more difficult skills that require the basic skills to accomplish.That is what I am currently trying to do for myself to improve the speed of my shooting. I am slow but getting faster or maybe I should say I'm getting less slow. Thanks for your response.