My primary shooting discipline over the years has been ISSF Standard Pistol. It's basically the same as American Bullseye, but a 60 shot match instead of a 30, 60, or 90 shot match. I've also shot a fair amount of American Bullseye. ISSF clubs are hard to find, but if there's one close, go for it.

Bullseye can be very intimidating, but 2700 matches that last all day, multiple guns (.45 and .22, at a minimum), 50 yard slow fire, etc. Ignore all that. Get a decent .22 pistol (an S&W Victory is fine, others prefer Buckmarks or Rugers; all work fine), a decent red dot, and a stack of targets. Possibly a trigger job, especially on a Ruger.

Find a local league if you can; they often shoot a 300 or 600 on a weekend evening. Shoot this with the .22, even if they have a centerfire match. If you decide to shoot a 2700, just sign up for the 900 portion in .22. If you feel ambitious, you can shoot the entire 2700 in .22, though only the .22 match will be for score.

But a good league is really the best bet. You'll get some good pointers from other shooters, and the environment is usually great with a really friendly group of shooters. It's like a bowling league, shooting with a little bit of social activity. Almost all the shooters are shooting against themselves, in that they want to get better.

Another way to go is get a basic competition air pistol, like a Gamo Compact and some 10 meter air pistol targets. Set up a range in your garage with a pellet trap made out of an electrical junction box and some plumbers putty. Go out start shooting 10 shot strings in the evening. Track your scores, calculate a handicap, and then shoot against one another with the handicap to see who does the dishes.