I use an air rifle for some pest control problems that I cannot solve through other methods. There is a lot of information and a multitude of choices on the market.
To begin, you'll need to determine a caliber. The basic premise is if it has feathers go with .177 caliber, if it has fur then you'll want a .22 caliber or larger.
Not all air rifles are equal, there is a lot of junk on the market. You basically get what you pay for when you are talking about air rifles. Your biggest concern is accuracy, you'll need to be able to put a pellet into a target that is about the size of a nickel at distance. Quality air rifles are very accurate and very effective at pest control.
There are several power systems to choose from: spring-piston, gas-piston, multi-stroke pneumatic, and PCP pneumatic. You can also get CO2 powered systems, but those products have accuracy issues as temperatures change.
My personal preference are the single stroke guns either spring piston or gas piston. I like quality products, so I'm not going to reccomend cheap junk. In reccomending a specific air rifle, stay away from the
Wal-Mart type of products. These are low end cheaply made products, some individual products perform accurately over tge short term, but they all fail in consistent quality and product durability. I went through the "let's just get a gun and go shooting" route...it can be fun, until things break, fall apart, or loose accuracy....so always go with quality and you will be happy over the long run.
My favorite air rifle is an Airarms TX200 in .22 caliber. The barrel has a built in suppressor, keeping the gun very quiet. The TX200 that I own lets me put a .22 caliber pellet into a circle the size of an aspirin tablet out to 60 yards.
My next choice of air rifle would be the RWS Diana...another great product.
On the low end (price wise) would be the Umarex Browning, at just under $200.00.
This is by no means an exhaustive list of reccomendations...I'm just mentioning the big hitters.
When choosing a scope, get quality optics rated for air rifles. Piston guns beat scopes to death...the physics of the recoil of an air rifles is in two apposing directions. Where a firearm has a recoil in a single direction.
For a scope I like the a Hawk Airmax EV 4-12x40. It's air rifle rated, and holds zero very well. It's a middle of the road product that I have found to be very reliable.
I hope this helps...