As mentioned, it is a fairly simple procedure that many people will need to get used to doing. I would suggest this being something you need to start practicing now rather than later. And so, for the present, since most people will probably can or freeze the meat for future use, at least until SHTF happens, the two most important things are to cool the meat as soon as possible, if keeping for any period of time, and it must be kept clean. These two things will prevent a severe taste problem, and reduce the concern of loosing meat to rotting and disease.
I think it is short-sighted not to consider the possibility of hunting larger game such as we have here in the US. At least for the immediate future there will be a good supply of large game that will be around for some time after SHTF happens. In some areas the game may begin to be hunted out after an extended period of time, but not necessarily everywhere. If you do take larger game, be prepared to process it for long term use so as not to waste any of the valuable meat. Unless you have a large group it will be difficult to consume it all very quickly before losing some of it. Larger game can be dressed and processed more easily if hung by its head.
I would suggest picking up a book on dressing and butchering large game animals or even cattle, which is very similar. Here is a great selection of books to add to your library on the subject including recipes. Don’t forget, some of the healthiest meat in the world is wild game. No chemical or growth hormones in the meat, and it’s low fat.
Totally agree with the OP, don’t waste the heart and liver. I love them with a pot of rice or in a soup. Some people like the kidneys so consider those also, but be sure to clean them thoroughly. He’s also right that the human body needs a certain amount of fat to function properly. As far as birds go, most are very small and have little meat anywhere but the breast. So, I usually only cut the breast off. On a larger bird such as turkey or geese, the legs should have enough meat to make it worth the trouble. If you have a problem with getting all the feathers off to suit you, you can just skin out the bird. The only problem with this is they tend to dry out more when cooking since the skin contains most of the fat and oils.
These books will also help you to purchase the right tools for the job. I would recommend a good meat saw, a couple a different knives, and game shears. Oh, and don’t forget a way to keep the blades sharp. A sharp knife is a safer knife, because it doesn’t require as much force to cut with and therefore less likely to slip and cut you. Hope this helps.