In your budget I'd buy a .22 cal. Weihrauch HW 95 from Air Guns of Arizona for $315. It's rated at 755fps which should be more than enough for your uses.

http://www.airgunsofarizona.com

Then I'd buy an RWS scope mount from Amazon for 26 bucks:

https://www.amazon.com/Umarex-23005...&sr=8-7&keywords=RWS+scope+mount

Then I'd buy a 10x SWFA mil-quad scope for $300. For about $650 you'd have an accurate, powerful rifle with a great scope that will allow you to spot your shots and hold over (or dial) for drop as well as close focusing. Pellets drop fast, and you will need a reticle with hold-over markings in order to place shots accurately past 20 yards or so.

Here's my setup for backyard pest-control: a Feinwerkbau 300S with an SWFA 12x scope in the RWS mount. This gun puts out about 650fps, but it is fantastically accurate, so I'm able to place shots precisely; for example, head shots on ground squirrels at 35 yards using the mil-quad reticle to hold over for the drop.

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I would stay away from the very high power break barrels that get advertised and promoted here in the US. A quality British or German air rifle may cost a bit more (though often they cost less) but will almost certainly have a better trigger, better sights, and better barrel than the big-box store break-barrels.

Last edited by Oregon45; 12/30/16.