Bunnies and squirrels... You need a .22, and it needs to be close to the legal limit of 12ft/lb - generally you won't get that with a really cheap air rifle.
Also
don't even think of going after bunnies or squirrels (or anything that isn't a paper target or a metal "knock down" target or that sort of thing) until you are a good enough shot to guarantee a kill on the first hit. Remember that the guaranteed kill zone on a rabbit or squirrel is generally considered to ONLY be a pellet TO THE HEAD.
Now, imagine the rabbit is 25-30 metres away. Are you certain that you can guarantee you'll get it in the head - i.e. through the brain - with the first shot?
Remember, a paper target doesn't feel pain and suffer if you put a pellet through the wrong bit.
A bit of advice I was given a while ago was that If you can bust an extra-strong mint at 35m every time, then you're almost good enough.
Also, you'll want a silencer on it so you don't spook other quarry with the noise.
Join a club, and practice, practice, practice!
I've just got back into airgunning myself. You may find this thread over on British Blades very helpful.
http://www.britishblades.com/forums/showthread.php?t=5829
As for which gun to get, well that depends on your budget! A reasonable guide can be found here
http://airgunbuyer.com/page.asp?pg=18. A good choice would be to join a club and talk to the members about what you want to do. Also a club is a good source of decent second hand rifles.
P.S. Read up on all the laws pertaining to air gun ownership and use e.g. if you're doing pest control (hunting) there is a limited amount of quarry you can legally hunt, and you need written permission from the landowner to be on their land shooting.