As a long time competitive shooter and Army firearms instructor, I can tell you that the most important factor in accurate handgun shooting is trigger control.
Stance, grip, sight picture and sight alignment are all vital and all can be taught to virtually anyone, but none of that means diddly if you can’t get the shot off while the sights are where they should be.
You can learn the basics through dry-fire. You need a smooth, uninterrupted trigger pull, the ability to call your shots and a consistent follow-through. Tape a 2” black disk to a blank wall. With an empty gun and no ammo in the room, stand 15’ away, get your stance and grip set up, get a good sight alignment/picture at 6:00 on the disk and concentrate on pulling the trigger straight back with no hesitation. When the hammer or striker falls, don’t move. Don’t blink. Where was your front sight in relation to the target? Practice this over and over. Call your shots and work on keeping the sight picture as close to a perfect 6:00 hold as possible.
No one can hold still, so don’t get frustrated when the gun moves. Strive for a “surprise break” and always keep your eyes open. A perfect hold isn’t necessary to a good group. The gun’s going to be moving. That’s to be expected. As you practice, the movement will be less and less.
When you take your gun to the range, put all this into practice. When the gun fires, it’s going to move, but you still need to know where that front sight was, which means you have to keep your eyes open. Try to hold the gun on target after it fires. You can’t, of course, but trying to do it is what I mean by “follow-through”. Call every shot, then check the target to see if you were right.
I shoot in a Thursday night pistol league, shooting my 9mm XD Mod.2 5” Tactical. My load produces a nice flash and I want to see my sights against that flash on every shot. If I don’t see the flash, that means I either blinked or wasn’t paying close enough attention.
My sight alignment/picture is almost never perfect, but it’s good enough that I can keep all my shots well inside the scoring rings.
If you’re going to hunt with your handgun, I strongly recommend that you get some experience in competitive shooting. Don’t worry if you think you’re not good enough. There are some pretty terrible pistol shooters in my league (and even on my team), but no one gives them a hard time. We do the best we can and help each other get better.
The experience of competition will get you used to shooting under pressure, which will help when you’re lining up on that deer.