Hunting small game, like rabbits, squirrels and birds is a sport. It is simultaniously a food gathering skill.
The techniques used for small game are also useful for larger game. Larger game requires a more powerful tool. I use rifles. A bow and arrow, or a crossbow, where legal, may also be used, although bows and crossbows require more practice with those particular tools. Traps and snares may also be used, although I have no more than a passing familiarity with them. My forte is a scoped rifle.
I've been using rifles for about 50 years. I'm not a great shot, but I am a good shot. Being a good shot is just a matter of a little practice, and perhaps a little instruction.
For small game, out to fifty yards, I prefer an air rifle, preferably in .20 and .22 caliber. I can generally hit the center of a dime out to about 30 yards. A good air rifle of sufficient accuracy and power may sell for about 4 or 5 hundred dollars, and a good scope may cost the same, more or less. the power range you need is between about 12 and 20 foot pounds of muzle energy. Other than pre-charged air rifles, the one cock spring powered air rifles are the most powerful, in the 12 to 20 foot pound range.
The best ammunition is lead alloy pellets. There are other materials, but lead alloy is generally the most practical, available and most accurate. The heavier pellets are generally more accurate than the lighter ones. The optimum weight for your particular rifle may be found with practice. Some pellets are pointed, some are flat nosed and some are rounded. I like the rounded pellets. Crossman Priemiere and Beeman are great pellets. You can't go wrong with either brand.
One of the beneficial characteristics of good quality air rifles is that they are virtually silent. You won't scare away other game, and if you miss, you can take your time sending another well aimed shot down range. Another benefit of air rifles is the cost of ammunition. Even the highest quality ammunition is half or even a third the price of gun powder ammo. Because of their low power and almost silent report it is possible to bractice with an air rifle unobtrusively in your back yard or even in your cellar. Safety, at all times is, of course, a primary concern. Outside, there should be no people or habitations for about one hundred yards beyond your target.
Practice with the same rifle and scope you are going to take to the woods with you. You need to feel comfortable with your rifle.
These are some of the basic mechanics and materials needed for small game hunting using an air rifle. Later on I'll discuss the woodsmanship, if you will, of small game hunting. Hunting is a walk in the woods, or in the mountains, or in the dessert. Go on a hunting trip armed with nothing more than a pair of binoculars, a compass, a map, a couple of sandwhiches, a thermos for hot coffee, a plastic bottle of water and perhaps a camera.