Be accurate
Every 10/22 I've ever bought, I've kept or given to one of my sons. So I'm not averse to semi-autos. But as a general rule, their triggers leave a bit (or much) to be desired and their accuracy is not target accuracy. Were I a squirrel, less than target accuracy in a human's rifle would not make me any bit bold.
I am prejudiced. I was on my high school rifle team and had a retired sgt. major on my @$$ all that time. So what do I recommend? Yep, a bolt action rifle with aperture sights. The rear aperture should NOT be the tiny target pin-hole when hunting or plinking. Screw-in the aperture with the more gracious eye opening. Better rifles often-times have better triggers. Read lots of rifle reviews -- but not ones from popular magazines, they're there to sell products.
Ammo: Use standard velocity or affordable target ammo. It's accurate. It kills small game without destroying the meat. You can use the CB cap ammo for here-to-there shots (10 yrds). If using CB caps, use the "long", so that the bullet doesn't jump to the rifling (destroying accuracy).
What's with this hyper-velocity ammo?! Who wants to blow a critter inside-out?! How you gonna eat that? If shooting dogs, a .22 magnum is minimum -- even then, we're talking head/neck shots on big ferals. Used-to, down South, feral dogs killed more deer than hunters killed deer. And then there's the matter of rabies outbreaks.