Hi Jack,
Im a keen shooter using both Airguns and Shotguns. You have to be over 17 to own an air rifle. You do not need a license for an air rifle or pistol as long the power is under 12Ft/lbs energy (rifle) or 6 ft/lbs (Pistol) - all guns you see in magazines online etc are within the limits.
Anything over that (powerwise) you need a firearms Cert for. Bloody nightmare to get and you might as well get a .22 rimfire if you get granted a FAC - as your only going to get 18 ft/lbs with an air rifle. .22 rimfire range for rabbits etc is max 100 yards, 75 yards for Foxes.
A .22 or .177 air rifle (calibre choice is up to you - .177 is flatter trajectory and faster (850 fps) but .22 has more impact as its a heavier pellet (600fps) ) - has a range of 30 yard for rabbits, pigeons and the like. Air pistols are only good for rats and close range (10 yards) if you get a full power one - there are only a couple that are full power. HW45 and Sheridan Benjamin for example.
I use .22 air rifle for rabbits,rats, sitting pheasant and .177 air rifle for roosting pigeons and magpies. Sometimes use a .20 air pistol for rats in the barns.
As far as shooting goes you need the permission of the landowner and all game season laws apply. Its also worth joining something like the BASC, which comes with shooting insurance.
Ferrets are cool and work well, but the cost of ferrets, collars, finders, nets etc can soon add up to the cost of a good budget air rifle and scope. You need to understand burrows and net every possible exit, also get ready to start digging when the ferret doesn't come out, because he / she has killed in the burrow. You also need to handle them daily and keep them occupied as they are nasty little sods when the get bored!
My wife is happy with me having a few guns but I'm not allowed ferrets!!
PM me if you want advise on buying a decent gun / scope combo within a certain budget.
Hope this helps!