Every once in awhile new shooters ask some basic shot gunning questions trying to decide which shotgun they should buy. They are generally concerned with either home defense, or hunting, but often both.
Yes you can hunt small game and game birds with the big 10 gauge, but it is more recoil than most people want to deal with on a regular basis. By and large the “big gun” is the 12 gauge. It has been a fight stopper from the OK corral to World War Two.
I have been asked many times if a 12 gauge is too much gun for hunting small game. There is a fear among some new shooters that the smaller animals will be too peppered with pellets to be edible or at best a gruesome mess. It is not true. Many, many, many rabbits and partridge and pheasants have gone down to the 12 gauge and been perfectly suitable for eating world wide over the last hundred years or so.
Sixteen gauge shotguns use a shell slightly smaller than the 12 gauge shell, but there is not a significant difference in the perceived recoil. In my opinion, the only thing gained by using a sixteen gauge is greater expense because the shells are more rare and thus cost more than the twelve gauge or the 20 gauge.
That said, a 20 gauge is also perfectly suitable for all shotgun hunting and for self defense without the full recoil from the larger shells. I agree with the often repeated advice that new shot-gunners, most women, and younger teens who wish to hunt or shoot trap or skeet should begin with a 20 gauge. Many are built for smaller frame shooters (youth models) and they will probably be a better match for those folks.
The 28 and 410 gauge shells are much smaller than the 20 gauge and also more expensive. They can be used for hunting and defense, but it is like deer hunting with a 22 rimfire. These shells are really not best suited to the job. These smaller gauges are useful for teaching shotgun use, and for youths to hunt squirrels with, but in my personal opinion they do not throw enough lead to reliably take birds on the wing. Others will disagree.
Once you decide on what shell to fire, the next question is what action choice to make for your shotgun. Single shots are simple to operate and inexpensive, but slower to reload than other types of actions. The venerable side by side shotguns aka “double barrels” are basically two single shot guns sharing a single stock. They are reliable in that you have two complete actions (triggers, hammers, chambers) so that if one breaks you still have the second, but they are also heavy. You are carrying two barrels. Pump guns are the next technological step. There are good ones and bad ones. They require two hands to operate and I have found them more prone to jam than any other action, but they did dominate the shotgun market for 50 years, so I really can’t put them down too much. Any of these can and will work for you if you find one that feels natural for you to operate. I personally love the old outside hammer side by sides but they are not the optimum for efficiency. In my opinion that designation goes to the more modern, semi automatic shotguns.
A semi auto will be more expensive than a pump, double, or single shot but it will kick less because the springs soak up recoil. Follow up shots will be very fast until you need to reload your tube or magazine. The number of shells you can load at a time varies but is generally at least three so that you will get at least one more shot than the old double guns without having to pump the action or work a bolt.
If you have any concern with recoil, get a semi auto. Regardless of what you buy, put a pad on the butt-stock to cushion your shoulder when you practice.
If you have a choice, buy a shotgun with a longer chamber to increase your ammo options, but buy the shorter shells to reduce recoil. You can fire shorter shells (like 2 3/4 inch length) in a shotgun that holds longer shells with no downside. You can't fire longer shells in a shotgun designed for shorter shells. For example you can not use 3 inch shells in a shotgun with a 2 3/4 inch chamber.
This covers the very basics of shot-gunning terms and should give you enough information to start asking questions and narrowing down the choices when you decide which shotgun you want. Above all be safe, and have fun.