In most states (we don't know what state you're in), you are required to have a hunting license to kill/hunt any animal. Also, most Fish and Game departments can seize your firearm/airgun, equipment and vehicle used to hunt an animal illegally. OT gave you some very good advice. P.S., he used to be in the business so he knows what he's talking about.

Update: Your IP address looks like you're in the Netherlands. I don't think either one of us have any idea about Netherlands game laws.

So going back to your first question of using an airgun, you'd probably want a .22 or .25 caliber for airgun hunting. A PCP would be ideal but again, I don't have any idea what the laws are like in the Netherlands.