Summarize:

If you're in an area where guns are tolerated, a .22 works well, with good accuracy. Typically, cottontail ("bunny") rabbits don't run away to escape; they "hide", often in plain sight behind the nearest bush, giving you a clear shot. Head shots are best, especially with anything more powerful than an airgun, as even a .22 will destroy meat, making rabbits more difficult to butcher. There are also ways to trap them. A crossbow bolt, BB, or a bullet through its brain works, as well as beheading it.
Rabbits, cottontails in particular, can be killed with an air rifle or small "toy" 50-lb crossbow. If you merely wound the animal, kill it as quickly and painlessly as possible.