Oldies but goodies,
Two that come to mind are the Winchester 69A and the old Remington Model 41 Targetmaster. Both can be easily had in your price range and are true American made classics.
The Winchester 69A came in a couple of flavors, the target model had a Lyman rear aperture target sight. It was a lower priced target gun that served as a plinker, or a higher priced plinker that served as an entry level smallbore target gun. All depended upon your perspective.
The single shot Remington Model 41 was the true boy's "first rifle" of it's time. I believe they sold for around $10 new. They were not on the market long, less than 5 years I think, but Remington sold a bunch of them and they're out there still. These rifles were incredibly accurate, the limiting factor being the lackluster sights. My Dad's is in my safe but still manages to get a couple squirrels menacing the birdfeeders every year. There's plenty of meat to drill and tap for a mounting a scope to wring out what the rifle is capable of, but that would be considered blasphemy in my case.
Last thought, back in the days of those rifles, run of the mill production 22s were typically not built of the highest quality steel. Should you come by and decide to purchase one of the older gents, if you stick to standard velocity or subsonic ammo it should shoot forever.
Take Care
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