>>1985633I can actually give decent info on this as a UK airgun retard (pic related).
You can hunt Pigeon, Rats, Rabbits and Squirrels (those are the top 4 anyway). To do this you need a permission, this is essentially an agreement between you and a landowner that they let you shoot and hunt there and that they trust you to shoot there, its not a formal document or anything but something in writing is generally considered good practice incase you are challenged (just a written note is fine).
Some general advice, practice before you go shooting (you can do this through hunter field target competitions or just static shooting) and do your homework, learn the permission you want to shoot on, where quarry nest or so on, know the times of year animals are out. There's plenty of good resources out there for you to discover on the subject.
A final note, check the power your guns putting out, a 6ftlb gun is not suitable for hunting, a 12ftlb (or thereabouts near the non FAC limit) is okay but you need reliable kill shots (typically the head).