if youre old enough to buy a handgun, i would look for a gun rental range in your area and see if they have any of the ones you want to rent.
if not, make sure you can handle/feel them before you buy.
most people dont like the P22 since its been sort of iffy. I have one and its been a good gun but it will jam from time to time depending on the ammo used. just so you know, alot of 22 autoloaders can be picky with ammo. sometimes you have to test various ammo to see what works best.
dont overlook revolvers. you can find some used S&W K22/M17 or M18, Colts Troopers and such. they are great revolvers.
you maybe able to find a used Ruger pistol. but most people know they are worth the $$$ even used.
i bought a single action heritage rough rider revolver a 6.5 inch barrel for about 200 dollars. its the best 22 i have even if it is off brand or however you would like to put it. they are great guns. i highly recommend them.
If you've little or no experience shooting a pistol then I'd recommend you go with a revolver instead of an automatic. Reason is that a beginner can pickup some really bad shooting habits if it becomes too easy to simply squeeze off another shot. A reasonable quality single action 22 revolver will give you really good service for alot of years without breaking the bank and it'll encourage you to think about what you are doing between shots. In addition, it makes cleanup alot easier than having to chase the empties from an automatic.
Depends a lot on intended use and also where you are located. For example, In PA where I live you cannot hunt with a Semiautomatic rifle or pistol. Shotguns are OK for small game but not deer or bear. A Browning Buckmark (which are good guns buy the way.) are only good for target or plinking. I chose a Ruger single 6 for that reason. You can get either a Ruger or Browning semi and be good to go. Good luck.
I'm basically a wheelgunner and l love revolvers. However, if you are just starting out I would suggest either the Browning Buck Mark, or the Ruger Mk. II or Mk. III. For one thing you can buy two good basic automatics for the price of a quality revolver. The Rugers are excellent little guns, but a lot of people have trouble reassembling them after they have been stripped down. But they are the best selling .22 automatic in the world. I've not owned a Buck Mark, but those I know that have them really like them. You can get either one under your $350 limit.