1) Don't shoot a rifle that kicks too hard or jumps too much. Either try another rifle in the same cartridge or shoot a different cartridge. This is a huge problem because people become entrenched in brands, names and labels and it distorts their ability to change. A guy with a rifle that doesn't behave properly will never shoot it as good as he could or to its potential.
2) Don't get wrapped up in a rifle. If it doesn't perform perfectly for you, alter it as many ways as you need to get it right. Who cares how much it cost or if it is "collectable". If you can't get it to work for you then get another rifle. If it is sentimental, hang it on the wall. If you want to collect rifles, don't shoot those. Too many guys hang on to a less than perfect rifle out of some nonsensical love affair with nostalgia, brand loyalty or sentiment. That less than perfect rifle always will be and so will their performance with it.
3) You cannot be over-gunned. This is a ridiculous term. If any good shot will work on the animal that is a good thing. You can be under-gunned. This is one of the hardest things for people to admit. It is not worth it to disappoint yourself and it is disrespectful to the animals. A 375 is a deer rifle when you are hunting deer with it, but a 243 is not an elk rifle when you are hunting elk with it. Ruining meat is a bunch of stuff. If you really need that extra one pound that may get torn then shoot a rabbit along with you big game or take a head shot. Even the eskimos who rely on every drop of meat don't even had a saying for ruining meat with a rifle. You can't eat it at all if it aint dead but you have to find it dead first.
4) Don't be afraid to take head shots. If you can shoot your rifle and it is as accurate as it should be you should be willing to take head shots out to whatever range you and the rifle can. This is really any range where the bullet is within an inch of where you are aiming. This is especially true if you are using a marginal caliber. When I culled bison out of Yellowstone in the '70's I used a 270. The group thought I was nuts, but I head shot them at 100-200 yards and mine all fell within a few steps of where I hit them. The body shot guys regardless of caliber had bison lying all over a one mile radius.
5) Don't accept marginal accuracy. Rifles could shoot three shots in to an inch and a half at 100 yards. I suppose this could have only gotten better. If your rifle can't do that you are cheating yourself when doing serious hunting. It might work for meat hunting but for trophy hunting I guarantee one day you will be disappointed because you lacked confidence to take a certain shot, can't take the shot because of the presentation or end up with a wounded animal.
6) A good scope with magnification makes it easier to hit precisely. If you don't have a particular eye problem, a scope is an advantage on standing game. Many times I've told a guy to shoot through a little opening in the brush or aim at a little wrinkle on the animal. The open sight guys can't see it or can't hit it. With that said, learn to shoot with sights. It will make you a better rifleman all around. If you have enough gun and are shooting still animals at mild range then sights will work, but I consider them meat hunting tools. If you are stuck on tradition you will have to make sure your personality is such that you are willing to pass up fully 65% of the opportunities you encounter. If you invest a lot of time, risk or money this can be detrimental to your success. Most personalities I've encountered are not prepared for an unsuccessful outing under such conditions, even the open sight guys that said they were were clearly disappointed, but it was their own fault.
7) Keep you rifle by your bed at night. First thing in the morning, shoulder the rifle, get a sight picture and work the action, three or four times. At night before turning in, repeat the same ritual. It is good for you and it is good for your rifle.
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9) Set up a .22 rifle just like your hunting rifle. Shoot small game with it, year round if you can. Shoot cans thrown on the ground and figure out the windage and elevation hold as you walk around them shooting at them. Shooting small game and shooting cans like this in field conditions is a much better teacher than shooting targets in a sterile environment. You are better off with one .22 and one similar big game rifle than a slew of big game rifles that are shot sparingly and used for a few weeks per year. If you don't train, you won't improve and you won't have the second nature to make shots when they are presented in the field. You many never forget how to ride the bicycle but if you don't ride every day and push your own limits you will never win the race.
10) Don't blame a rifle or gun for what humans do with them. Never take an unclear shot or one that risks human safety or a working animal like a dog or horse. Guns go bang when someone pulls the trigger. That should be intentional and with conscious thought of where the bullet will go. No animal or stunt is worth the consequense of a greedy, egotistic or thoughtless pull of the trigger.