The S&W #41, and the target grade High Standards are excellent guns, and are out there used in your price range, if you look around some. If you want to buy new, the Ruger Mk lll target version is a very good performer. I own all three, and none are bad choices. You should have enough & left in your budget for a trigger job, if needed, and maybe a red dot sight if allowed in the type of matches you are shooting.
Larry
Larry gives you some excellent advice.
I'm assuming your getting involved with the NRA Bullseye shooting? Its a fun and very good sport to hone your shooting skills. I have been shooting "bullseye" for over 25 years now.
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I have shot all of the guns mentioned above and quite a few more! (I'm also a pistol instructor and carry a master classification with a pistol.)
My advice mirrors Larry's.
Really, you don't spend too much now, buy used and spend the extra on a trigger job and lots a ammo. Later, as you improve you can trade up to a better gun. A gun doesn't make the shooter, the shooter makes the gun shoot, believe me! Now, having said that, once you get shooting form muscle memory you will benefit from a ergonomically correct firearm like a Pardini, Benelli, Hammerli or upper end Walther. Now these will drain the bank account!! They average close to 2K.
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These will solidify your targets and bring up your averages.
The S&W is a fine pistol, but overpriced and heavy. Yea, I have owned two of them and it will work well, but for the $$ there are better choices.
Ones I will recommend you steer clear of are the cheap S&W 22's 22A and its ilk. The Beretta Neos. The Walther plastic black and white jobber and The Sig Trail side. no offence to those owning these. they can be great plinkers but bullseye guns they are not.
If I where pressed to recommend one or two it would be the Ruger or the Browning and would lean to the Ruger simply because the after market embraces it with hop up parts.
A red dot is a good thing to be competitive. But IMHO you are far better off shooting and learning your iron sites for a couple seasons. Even tho your average will go up with the dot. You don't build a house form the top down, you need a foundation. That foundation is built learning to use iron sites.
Good luck,
CW