Well, I'm going to really go against the grain here and recommend a .357 with a 4" barrel.
The 2 and 3 inch barrels are still hard to shoot accurately. I've been using wheel guns since I was a kid and I still have a hard time with the short barrels. A 4" is about the perfect length for balance and accuracy.
The revolver can shoot wax bullets, which run $25 for a thousand. Load the 'bullet', cap with a primer and you're good to go. Really cheap to shoot and lots of fun. Then, you can switch over to very low power, 38 special loads with almost no recoil and go up from there. You can load a gun like this for everything from mild to 'OMG!'.
The .22's you're looking at will run you around 400 dollars and for that much, or a hundred more, you can get a Ruger Speed or Security Six, SP 101, S&W Mod 19 (what I use), or a S&W 586/686....all used prices, but there's lots of used revolvers sitting in gun store cases.
And, learning to shoot a revolver accurately means learning good trigger control. That transfers over very well to the semi-autos with their lighter triggers. Revolvers are very safe with their heavy trigger pulls and simple operation.
Good Luck in the gun hunt!