Gamo Wildcat Whisper in .177 cal
They shoot 1350fps and are deadly up to about 60yds and cost about $120
The triggers on Gamos aren't very good, but they can be upgraded. A poor trigger on a break action airgun is a good way to ensure you miss your target due to the double recoil of the way spring and gas piston-powered airguns operate. A good trigger will have a pull weight of less than 2lb and have a predictable second stage with very little creep. A good trigger breaks like a piece of glass when it engages. The 1350FPS is a marketing tactic, they use lead free ammo (light weight) to obtain those numbers from a chronograph. With airguns, you don't want the ammo to go supersonic because pellets become unstable at that point. Instead of rotating like a bullet, they tumble and go off target. Ideally, you want something between 700 and 1000FPS.
A good all around caliber is .22, you get great velocity, little arch in trajectory, and a lot of knockdown power. Smaller calibers lose more energy over longer distances while larger calibers deliver more energy at the expense of a flatter trajectory. The big exception to this rule is a pre-charged (PCP) rifle, which use a cylinder of compressed air and a magazine. You can dial up (or down) the air pressure to get more power or more shots per fill. PCP guns can be very powerful, for instance, a Quackenbush PCP rifle in .50 is enough to take down a buffalo at 200yd and be quiet as a mouse fart.
My go to pest rifle is this guy
http://flyingdragonairrifles.org/index.php?route=product/product&product_id=51 outfitted with a globe and post up front and a Williams peep sight in the back. Peep sights are ace for shooting under 50 yard due to the super fast target acquisition compared to a scope. I highly suggest the fully tuned variant, comes with a killer 2 stage adjustable trigger, recrowned barrel, honed compression chamber, fully lubed with Honda Moly60, deburred internals, and a urethane piston seal. Comes with a transferable lifetime warranty too. That gun will take out anything up to possums and raccoons out to 50yd or so.
If you want to get into PCP cheap, get a Benjamin Discovery
http://www.pyramydair.com/s/m/Benjamin_Discovery_Rifle_Pump/1534 and a TKO Muzzle Brake
http://www.tko22.com/ That will be more powerful than most break action air rifles at that price point and deliver more energy down range and better accuracy due to it not having double recoil like a spring gun. Make sure you get that muzzle brake, or else you'll arouse suspicion in your neighbors. It'll sound like a sneeze with the brake, the pellet hitting the target, even a soft bodied critter, will be louder than the report from the muzzle.
Unless you truly like the thrill of the hunt and then a slingshot with a pinch full of BB's is economical. 
