3 answers:
the 10/22 is a good basic all around high quality midrange rifle. It's not too fussy about ammo
on the cheaper end the basic Savage .22 auto is *fine*. totally fine.
But I'd go get a used one. If you don't mind the weight of a full sized rifle, and feel comfortable checking for pitting with a flashlight, the Springfield model 87 (M1938 training rifle, Sears Ranger 101-16, also made by other manufacturers) is the bomb.
16 round tube magazine for .22LR in semi-auto, will fire long or short (or LR) as a manual bolt repeater or single shot. Dead simple to fix, clean, and maintain. Accurate, solid, well balanced. It's a 24 inch barrel, this isn't a carbine by any stretch, but it's a Real Hunting .22
There *are* scope mounts, but they can be hard to find. I don't need one for squirrel and larger game at any reasonable range.
Get the 10/22 or the less expensive current model savage repeater, but if you have gun shows locally and want a $60-$80 gun......
Ammo:
I do like heavy subsonics for short range but I have had good results all around with the CCI mini-mag HP and, honestly, I've never had a problem with the winchester Xpert22 cheapo ammo downing animals.