Small game such as rabbits, squirrels, etc. are fair game for the rimfire .22. You wouldn't be able to legally hunt deer with a rimfire .22, however. The .22 is a great starter rifle, gets you used to the mechanics of shooting, is inexpensive, and is fun to shoot because it is about as light as you can go power-wise in a rifle. It will be limited to smaller game as a hunting rifle, however. If you plan on taking anything larger at some time, like deer or hogs, for instance, or if you want to shoot at long range (over @ 100 yards) you might consider a more powerful centerfire rifle.
Most folks I know have the .22 they started with and at least one second more powerful rifle for larger game. But since you are starting as an adult, you might want to skip over the .22 in favor of a light centerfire capable of taking smaller and larger game with different types of ammunition. .22-250, .222, .223, and a handful of other light centerfire cartridges can do double duty on small game and deer or hogs. They can also reach out to much greater distances with better accuracy than the .22 rimfire.
