I'm going to add my 22 cents worth.
With the way things are going with the liberal gun bans, you should concider this.
Do you think you will ever want a semi auto rim fire rifle? If so maybe you should start with a Ruger 10/22. This will get your foot in the door to the semis, Now I can't garantee that the government wont ban these firearms rendering your 10/22 a safe queen which you may have to surrender to the government later down the road. But at least you would have been able to experience the semi world. The Ruger 10/22 is a great gun for plinking as it is light and easy to carry if you choose to go hunting small animals. It also makes for a great choice if you wish to get into the rimfire steel change game. (red dot if you want to play in the optics division) The Ruger 10/22 is also the Chevy 350 of the rimfire world with a great variety of aftermarket accessories should you wish to start customizing your firearm. The downside of the Ruger 10/22 or any semi for that matter is that you have to be very mindful of the fact that it is instantly ready for the next shot. Muzzle control and trigger finger control are paramount. But that being said must be diligent with muzzle and trigger finger control with all firearms, it's just that semis are less forgiving if you should make a mistake.
With that all being said you have to seriously ask yourself, are you willing to take a chance on a semi auto rimfire? If so the Ruger 10/22 would definately fit the bill.
If you are looking at precision rimfire the Bolt choices mentioned by other are great. I personally hav a CZ457 MTR with a fluted match barrel mounted in an MTD chassis with a Vortex Strike Eagle 5-25 x 56 scope . I've about 3K into this rig and it is great for precision shooting. This is one heavy rig and it would be totally useless in a rimfire steel challenge. Too heavy and too slow. Don't get me wrong. I love this gun, for what it is built for.
Someone mentioned a lever rimfire. I don't perrsonally own one (yet) but if I were to get one I would seriously look at the Browning BLR22. They carry a hefty price tag but I realy like the short thow lever action and athey are a very pretty looking gun in you go into their upper end variants
I suggest thaat you go out to the range and talk to some people and ask them what their thoughts are, I'm willing to bet that you will get a lot of different opinions and if you are luck some of these people will offer up their firearms for you to try. The world is full of important choices.
Good luck with your decision