One very important visual skill the red dot pistol shooter needs is to learn the "Visual Hand Off". This is a term we use to describe what happens with the eyes as a shooter new to the red dot system is learning to use it.
On a rifle, the head and eye is automatically positioned to pick up that red dot as the rifle is brought into the shoulder. A pistol however, floats in space, held there by the two hands. There is no third and fourth point of contact to place the eye correctly. Regardless of the uniformity of your draw, if your eye is not in the visual cone of the red dot, you will not pick it up...nor the sights for that matter.
The way to solve this is with the use of the back up iron sights, and we insist that they need to be placed in the traditional positions on the slide. It is best to look for the iron sights FIRST. Especially with contorted field shooting positions or from positions required for using cover - in other words "other than standard" shooting positions.
Most shooters find that going to the rear sight happens first, even if they try to seek that front sight first. I suspect, with older shooters, it has to do with the reality that the rear sight is closer and easier to see. Then as they bring rear sight and front sight into alignments (something that happens very quickly because they have done it countless times in the past), this gets their eyes in the right place. The instant the sights come into view, the red dot is evident. This is the moment of "Hand Off".
The eyes - and brain actually - shift from seeking iron sight alignment to accepting the presence of the red dot, and abandoning any further concern with the iron sights. The dot, being in the infinite focal plane is placed on target and the shot will land where that red dot appears.
Now the issue comes up about whether it is best to focus on the dot or the threat. My answer is the usual...it depends. For targets inside about 20 to 25 yards, I find I can keep visual focus on the target and notice the dot on that target. For a large target that is very sufficient. I find this gives me the greatest speed of shots. Now that distance may be more or less for the reader as it does depend on the individual eyes.
If the shot is challenging, or the target distant, I visually focus on the red dot itself as if it was a front sight and press. That red dot focus seems to give me the greatest degree of accuracy.
Once you own the "hand off", you will find your speed and accuracy increase dramatically.