Some of the newer air rifles are as fast in FPS as .22's at close ranges...and can be just as deadly.
Care in shot placement and ensuring your backstop are highly important. Safety first!
That being said, I'd chat with the neighbors before buying anything...just to find out where they stand on the varmint issue.
Last thing you wanna have is a PETA Nut living next door, while you're popping squirrels & whatnot out of the trees.
Bad Juju. Very Bad Juju. Whether with an air gun, slingshot, or a .22lr
Then ya check local laws...like Vincine sez
Once all that's resolved...then you start figuring out whether you are a pistol guy or a rifle guy.
Some folks are naturally better at one or the other...or have had more training...which makes a difference.
Then with the info from the neighbors/legal restrictions learned, you then pick air-gun or firearm.
If you want to be REALLY nice, and everything is in agreement with firearms, PUT A CAN ON IT!
$200 tax stamp is WELL worth the benefits of less firearm noise when dealing with neighbors...
My fav pistol for close-up snake/squirrel/rat squishing is a High Standard model B with Aguila subsonic loads.
Nice & quiet, & High Standard is well known for VERY precise pistols, they beat heck outta Ruger Marks & Browning Buckmarks.
I've got a Marlin 925M (.22WMR) bolt-action with scope for long-range rabbit/coyote/other varmint work in my gardens...
up to 150 yards of KZ covered completely