Bolt guns
You bench rests 22LRs and measure your groups and somethings come to light. Quality bolt guns always out shoot the out of box autos. The auto action is usually rattling around in the stock and the thin barrels are vibrating with all the commotion. I spent about 50 hours trying to get a stock 10/22 to shoot. I got it from 2" down to a reliable 3/4 inch at 50. All of the stock autos I have tested shot about 1.5-2" at 50 regardless of ammo.
The modified Ruger 10/22 with a quality .990 bull barrel, neutral bedding and quality trigger can really shoot. I have seen one with a Lija barrel shoot with or better than some Anschutzes in competion. Probably cause the owner was a good shooter who could read the wind and was willing to sort lots of very expensive ammo Most people don't put a $350 barrel on a 10/22 and bed it.
If you want an accurate .22LR, go with a heavy barrel bolt gun with a good trigger. I bought a pair a Savage MKII heavy barrels just to try them. They have cut riflling and match chambers at a very low price. The barrels are not screwed into the receivers so I thought they wouln't shoot that great. Wrong. They shoot with the purpose built target guns and they are suprising un-picky about ammo. Specific lots of Eley shoot 1/4 inch at 50. Under 1/2" with boreing regularity. Both rifles do better than the .22LR bullet in a light breeze. You might consider one. They now have the excellent accutrigger.
I really recommend a good quality 3-9 or 4-12 scope you can set the parallex at 50 yards for target shooting. For a squirrel rifle, I like the 2-7 or 3-9 quality rimfire scopes. The parallex is set at 50 yards rather than 100-200 yards. Once you have good rifle and scope, ammo becomes the deciding factor for rimfire. The economy ammo, std high velocity copper washed, and regular target ammo varies wildly in each gun and most people wont feed their rifle much $10-$25 ammo.
Remember, with a .22LR, you sort through lots of ammo to find lots that work in your chamber. Even ultra expensive target ammo can shoot very poorly in some barrels (there are very good reasons for this do to bullet upset and size). You find out which high quality target ammo (Eley, Gold Metal, RWS, etc..) your gun likes and then which lots are the standouts. I have my own 1/2" outline circle targets in 2 rows of 6 on a piece of copier paper. IN ZERO WIND with a full set of bags and very best shooting techniques, I shoot a group with each lot of ammo with two rifles side by side (example: left column Savage MK II, right column custom barrel bolt gun and each row a different lot). It is amazing how one lot will do poorly in one rifle, but really good in another. When you find a standout lot, you buy more. With the very expensive ammo, I don't like to buy too many lots to try. I usually buy one lot, find the gun that likes it best and use it in that gun. If it really shoots, I buy in a good supply. I also sort lots of regular high velocity bulk ammo this way and some std target. Some lots shoot surprizing well in a particular barrel as long as it stays supersonic to the target. No rythm or reason to it. When I find a really good lot, I buy a bunch of it. The surplus bad ammo get run through the pistol or my kid shoots it up.
Probably more than you wanted to read. Buy a high quality heavy barrel bolt gun with a good trigger. The Savage it very economical and has the barrel quality and trigger you need. Put a quality scope on it ($75-120 is usually good enough), and test and sort ammo to find the ammo it likes. Watch the wind! Sorting and testing ammo will make you a much better shooter and that is the great part about serious rim fire shooting. I think you will be very satisfied with the results espcially when you start comparing targets with the "plinkers" at the range.