Griz, one thing to be aware of is that if you are new to handling and shooting handguns, you will be teaching yourself some new nerve/muscle motor reflex skills. Much good advice stated above, but realize that point-shooting takes way more practice than aimed shooting, and that successful point shooting requires a very thoroughly programmed connection between your eye, your brain, and the muscles and nerves of your hand and arm. Lots of reps required to get there. As you raise the pistol from the holster to firing position, this is called the 'presentation' of the pistol. This motion is what will eventually become what some call instinct, but is actually programmed learned reflex.
I would suggest starting out with aimed fire. As someone stated above, focus on the spot you want to hit, bring the pistol sights in line with your eyesight so that the sights go to that spot, then CHANGE THE FOCUS OF YOUR EYES TO THE FRONT SIGHT TOP SURFACE. Don't shoot with blurry sights, always see and focus on the top of the front sight and get as perfect a front/rear sight alignment as possible. Press the trigger gently straight rearwards without disturbing the gun, and try to make the exact moment of discharge a surprise. This will take a delay of a few seconds but with sufficient (a lot) practice can still be a surprise at only a fraction of a second. This will prevent flinching - pushing the barrel down in anticipation of recoil before the shot is fired.
This act of presentation to eyesight line helps teach your hands to bring the pistol in line with what your eye is looking at. Seeing the sights VERIFIES to your mind that the pistol is indeed aligned correctly (something you can't check in 'point' shooting). Use presentation to sight use to program your hands. Enough practice with this, and you'll not only be a good sight shooter, but your presentation will be more and more perfect as the gun comes up, requiring less correction, and eventually will be nearly perfect instantly (as verified by sight usage).
At this point, you will find you can hit at closer ranges on big targets merely by presenting the gun and looking over the top of it, since your HANDS will be providing near-perfect alignment as a trained reflex. When you get to this point, you can still hit even with the gun below your sight line but within your peripheral range of vision (i.e. mid-chest height and halfway extended). Range time will tell you what your distance/target size limitations are. As you get better and better with more sight usage practice, 'point' shooting will get better as your motor reflexes are honed.
My main point here is that by practicing point shooting first, you will have no idea where the gun is pointing until you see the hole in the target, which frankly does little or nothing to train your hand or hands to get the gun in alignment with what you are looking at, since you don't have a reference while you are pointing the gun (unless you use a laser). So, you will burn a lot of ammo with little or no feedback to make you better. Use the verification of sight picture to train your mind and hands first, then you will have something to work with for 'point' shooting up close.
Karl, sounds like you've had something to do with Col. Cooper...