I'll quote myself here and I suggested the Cometa based on your "about $500" budget. You'd get a rifle completely gone through by Jon to make sure everything is good to go, still have money left to put towards a decent scope and pellets, plus get a rifle with the Lifetime Warranty he mentioned above.$500 would get you the Cometa gone through by Jon, plus a decent springer rated scope and pellets to get you started.
Caliber would be your choice, but either .177 or .22 would easily fill both target and pesting duties for squirrels, rabbits and whatever pest birds you might want rid of.
While I can't argue with the HW95 mentioned by fellow member MyCapt up there owning one in .22 myself along with it's cousin, a Beeman R9 .20, you'd have to up your budget quite a bit to get a new one. You might well find a good used one surfing through Ebay or posting a Want to Buy ad here once you get the forum minimum posting requirements.
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