I personally own a Ruger 10/22 (my first rifle), a Marlin 60, and a Savage Mark II.
The Ruger and the Marlin are semiauto. The Marlin is MUCH more accurate in unmodified form. The Ruger CAN be highly customized to make it more accurate than the Marlin, but the Marlin starts out as the more accurate and more affordable rifle. The Marlin has a higher ammo capacity in its tube magazine than the Ruger does in its factory rotary magazine; however, the Marlin's tube magazine is not detachable. For small game hunting, this would not bother me, since marlin 60s come with a magazine capacity of 14 to 18 rounds, depending on how old they are. If you can't stop to reload after hitting (or missing) that many squirrel or rabbits, you should review your hunting habits.
The Savage Mark II is also VERY affordable and VERY accurate without any modifications, but it is a bolt action rifle. You may not get as high a rate of fire out of it, but you really should not be doing any rapid-fire while hunting for small game. Mine came with a 10 round magazine from the factory, so it at least meets that requirement. The thiing I really love about my Savage is that it isn't picky about ammo. Semiautomatic .22 rifles are notorious for being perfectly reliable with one type of ammo and awful jam-machines with others. A bolt or lever action rifle will eat whatever ammo you feed it, since the action is cycled manually and not dependent upon the ammo.
For the use you described, I'm guessing what you really want is a .22LR rifle or possibly one of the .17 caliber rimfires. You mentioned in your original post that .223 would be okay, but .223 is a very very different round from .22LR.
.223 Remington and .22 Hornet use bullets that are roughly .22 caliber in diameter, but they are centerfire rounds with a lot more power than typical rimfire cartridges used in small game hunting. The sporting use of .22 centerfire cartridges is generally varmint hunting (coyotes, groundhogs, etc). It is technically legal to use them on deer in some jurisdictions, but they are not considered ideal deer cartridges by many hunters. They are probably overkill for squirrels and rabbits.
.22LR is cheap enough to practice with A LOT. It is also up to the task of taking small game. Other rimfire cartridges that are common for small game actually use smaller diameter bullets that go much faster for a flatter trajectory than .22LR. You may have heard of .17HMR or .17HM2. These may offer some advantages in terms of long range accuracy, but they also cost much more than .22LR ammo.
.22WMR (commonly called .22 Magnum) is another common rimfire round. It looks like a longer version of .22LR, and it packs a slightly higher punch. Once again, though, it's more expensive to practice with than .22LR.
Final verdict: A good rifle chambered for .22LR will do what you want cheaply enough and accurately enough.