The first thing you must understand is that all pistol rounds are a compromise. Second, shot placement trumps 'stopping power' all day long. You must decide what's most important to you.
Generally speaking, the 9mm is the most versatile round to start with. Readily available, reasonable price, high capacity, limited recoil and plenty of styles and options for all of your shooting needs.
The .380 is basically a 9mm short. This means a comparable projectile but with a smaller case therefore less powder. Less powder means less recoil. Many 'pocket' guns come chambered in .380. With so many single stack, small 9mm - I don't see the advantage of the .380.
The .45ACP is a much larger projectile with fairly moderate recoil. As such; you lose capacity and the rounds are much more expensive. Many of your 1911's are chambered in .45ACP.
The 10mm was designed to have the mass of a .45ACP but the capacity and velocity of a 9mm. Bigger cartridge means more recoil and larger framed guns that can be hard for some to hold.
The .40S&W is basically a 10mm short. It's a very 'snappy' cartridge. if you are recoil sensitive, you should stay away from the .40S&W. That larger projectile does come at an extra cost per round.
The .22 has many iterations. .22LR (long rifle) is the most common. It is a rimfire case which means the primer that ignites the powder is in the rim as opposed to a center fire cartridge. All of the other ammunition types discussed have been center fire which means there is a visible primer in the center of the casing.
The .22LR is the least expensive and least recoil. It is an excellent step up from dry practice to grow your skills. It does suffer in the reliability category and it is a very small projectile.
This is not a definitive list. We did not cover typical revolver ammunition or some of the more obscure ammunition types. These are what we consider the most common and practical for fundamental shooting.
Plinking rounds are less expensive and, normally, a round nosed bullet as opposed to a hollow point. These can be loose packed or in trays of 50. Plinking rounds are an excellent way to practice, get used to recoil and learning how to operate your pistol.
Self Defense rounds tend to be expensive, generally are hollow points and are typically come in 20 or 25 round boxes.
It is never recommended to use re-manufactured or hand loaded ammunition in a self defense situation.
Self defense rounds should be from a reputable manufacturer such as Federal, Hornady or Barnes.
Whatever you decide to run in your firearms 1> ensure that you are using the correct type (don't put +P in a non-+P pistol - consult your owners guide or a competent gun smith) and 2> that you run a full magazine through your gun to ensure it feeds correctly.
We also recommend that you replace your self defense ammunition annually.