Diana New to the game

$500 would get you the Cometa gone through by Jon, plus a decent springer rated scope and pellets to get you started.
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.
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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If you want next level fit and finish and ludicrous accuracy the FWB Sport from Champions Choice is the way to go, You may need a new spring guide that fits better the the stock one.

There's still one left:

https://www.champchoice.com/used-fwb-mod-sport-break-barrel-air-rifle-18-joule-fwb0297

R9/HW95 in either .177 or .22 are terrific choices.

I have all three mentioned, and a couple I can't even look at without tinge of embarrassment :whistle:

I'm thinking about a Cometa!
 
Around 500
Find a used TX200 or Weirauch 97. Great triggers and superb accuracy. Never understood when you want to get involved with an activity starting with low end equipment, seems like a good way to get discouraged. I have a Tx, D54, and 2 D75s. 3 of them bought used, no regrets and all shoot dimes at 25 yards on a regular basis.
 
Got a .22 cal
5 at 25 with my HW95 .22.
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Outdoors a .22 is all over a .177 in my experience. All 3 of my .177s like the same pellet, JSB 8.44 in .452. However, in all brands the fact they list a headsize is pretty much a joke. I no longer weigh or size. Wife and I shot outdoors today 2 .177s. Tx200 and a Diana75 with a 5 mph switching wind. When you shot before the flag switched you could shoot in a quarter but it was hard to string shots together. At 25 yards no problem shooting under a nickel. 50 was a challenge.
 
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