My suggestion in approximately that price range is the Ruger Mark II or III. I vastly prefer the bull barrel. The Ruger 22/45 is excellent too with the bull barrel.
The Ruger SR-22P is a good little 22 pistol. Fits your budget. It's not a hunting gun or serious target gun, but a great plinking gun. I am really pleased with mine. But I already have a Mark II and quite a few other ones. So a plinker is fun to have. One needs to remember that the dominant use of most any 22 handgun is plinking.
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I've been eying the Heritage Rough Rider because it's pretty inexpensive
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That's one I don't have. If you are thinking long term, I would go with a more expensive Ruger Single Six. You will be happier in the long run and won't want to replace it like you probably will the Rough Rider in a couple years after you really start enjoying shooting 22's.