***Disclaimer - the narrative below is for defensive pistol shooting under duress to achieve speed and accuracy. Its focus is not range accuracy at fixed distance and fixed shooter position with unlimited time between shots.
To achieve accuracy with a defensive handgun, stance, dominant hand grip, support hand grip, arm/forearm position are all foundational. Grip depends on pistol type and hand size. There are different grips for semi auto vs revolver shooting.
Finger pad position on the trigger is also important. The middle of the pad of your finger tip should be centered on your trigger, not the joint between 2nd and 3rd phalanx bone. With your finger pad centered, when you make the shot, depress the trigger slowly straight back toward you. Pull that finger pad toward you and hold it there through the shot. A malpostioned finger or finger joint on the trigger will often result in a shot pulled low and away from the target.
Proper trigger finger positioning and trigger pull allow you to begin to understand trigger reset. After the shot slowly allow your finger to drift forward and you will feel the click of the trigger reset. Stop your finger right there and you are ready to press into the next shot. This allows you to move your finger and the trigger a shorter distance between shots, adding improved feel and accuracy. Trigger reset and trigger pull vary widely depending on pistol action and type.
You mentioned 1911 shooting, which are SA semi autos classically with amazing triggers. They typically have short resets and crisp, clean trigger breaks that are fantastic when compared to most out of the box striker fired pistols or DA revolvers or DA/SA semis so you've likely got a pretty good trigger on your 1911.
The paragraph above focuses on trigger portion of handgun shooting, but sight alignment and sight picture are the other parts of the equation. And these 2 parts of the shooting equation work together but are built on a foundation of stance, grip (both hands are key) and arm position.
You want to make your shot with both arms in extension, with no bend in your elbows. Certainly not the historically taught stance and grip with a slight bend in your dominant elbow, body angled to the side, support hand under the grip supporting the gun like a teacup. This is how I learned to shoot when I was a kid - it was not ideal. I had to break all that down and build a more consistent and stable foundation from scratch.
Arms should work together to create an A shape. Torso faces your target. No angle to body or if any, just slightly so. Weight is on the center of your feet, slight forward bend to your back at your hips. Maybe 60-70% of your weight on the balls of your feet. Think of an athletic stance but not with your heels off the ground. Slight bend in the knees. This allows you to absorb recoil and also move more easily when shooting when you start doing this as a component of defensive pistol training. These, when combined with stance and grip form the foundation of a solid shooting position.
Support hand needs to be on the gun, on as much of the gun as possible. For a semi auto, this means that the support hand fingers are positioned around the front of the grip, with the support thumb and thumb pad canted forward, almost uncomfortably so, toward the muzzle, so that the support thumb and thumb base are running along the side of the slide , in front of and to support the dominant hand thumb. The support thumb and thumb base can and should be pressed there for additional support. You then use pressure from your dominant hand index finger base knuckle and support hand thumb base to squeeze or sandwich the pistol between your two hands, just below the slide. If you will work on this support hand grip it will make a huge difference.
It is from this foundation that the trigger press and sight picture work together to make accurate (and eventually fast) shot placement.
I haven't talked much about sight picture as it varies based on iron sights or red dots. But the bottom line is that you must make sure you have a good sight picture before you initiate the trigger press of the next shot. Don't start the trigger pull until you see your target clearly and have also aligned your sights. This is where red dots really shine once you learn how to get them on target as target ID and sight alignment happen simultaneously. When that shot recoils, you must again regain your sight picture before making the next shot.
***Disclaimer 2 - I am by no means a professional shooting instructor, just a guy who geeked out on defensive pistol shooting for a number of years. A pro may come along and correct some of this and if he/she does, by all means listen to them and ignore my advice.
I have taken a decent number of defensive pistol shooting classes from KR training. These are principles I learned over my time working with them. Can't say enough about that group, really made me a competent pistol shooter. Bender Rodriguez on this forum also is a great instructor and resource and he will gladly give you a lesson.
The next thing (after you can achieve accuracy goals at fixed distance and stance) is to introduce the stress of time with a shot timer or even better yet competition. Bender is the best advocate on this forum for stressing yourself with at minimum a shot timer in your practice sessions or even better yet, competition, as those skills learned when competing are the closet you will ever get to real life defensive shooting.
