From the direction this discussion has been going, can we assume you [pricelessppp] are looking for a rifle rather than a handgun? Your original post didn't specify.
Assuming that you are looking for a rifle, so far we have suggestions for a Ruger 10/22, and various suggestions for ways to get into the AR-15 platform. Both will get you shooting, but they have some differences.
The Ruger 10/22 is a .22 LR firearm. You can, if you choose, spend a LOT of money after the initial purchase gussying it up with bells and whistles and "tactical" looking stocks but, after you've spent all that money, you still have a .22 LR Ruger rifle. But, for a first rifle as a "plinker," the Ruger 10/22 is arguably THE most popular choice in the United States.
Switching to the AR-15, now you've really opened a can of worms. First off, do you want .22 LR or should you go .223/5.56? You mentioned plinking. "Plinking" is fun shooting, and the more you shoot the more fun you have. Look at ammo prices. .22 LR runs from around 3 cents per round in bulk to maybe 12 or 15 cents per round if you buy the "good stuff" in 50-round boxes.
.223/5.56 is centerfire rifle ammo. It's in a different universe from .22 LR. Looking at one site, I found .223/5.56 in 1000-round lots for $341 (plus shipping). That's 34 cents a round -- compared to 3 cents per round for .22 LR. Don't want to spend over $300 for you first batch of ammo? .223/5.56 at that same web site is $7.50 to $8.00 per box of 20. Call it $7.75, and that works out to almost 39 cents per round.
Next: You can buy some dedicated .22 LR firearms that look like AR-15s, but they can't be switched over to fire another caliber/cartridge. Or you can buy a true AR-15 with a .223/5.56 upper on it, or with a .22 LR upper on it, and then when you want to change you can just buy another upper.
You can do that. However, I think I'm on solid ground in saying that probably most shooters with AR-15s don't switch uppers -- they just buy (or build) another rifle. There's just something about a lonely upper sitting there that cries out, "I feel so abandoned. Please make me complete again!" And, unless you have more willpower than most of us, you'll succumb to that siren song.
If you're just starting, as the Red King said to Alice in Through the Looking Glass, "It is best to begin at the beginning." Open/iron sights, no optics, a simple rifle that's easy to shoot and easy to clean and maintain. And that's affordable to shoot. .223/5.56 ammo is not plinking ammo. It's "I want to shoot something a ways off" ammo -- like at 100 yards, 200 yards, 500 yards. That's not typical plinking distance, that's serious shooting.
So I'll don my certified olde curmudgeon hat and suggest that you skip .223/5.56 until you have developed as a shooter. Another point is that most Appleseed shoots are at 25 meters. You CAN shoot .223/5.56 at an Appleseed, but you're probably going to go through at least 100 rounds (maybe more -- I don't remember). If it's 100 rounds, that's $15 for .22 LR compared to $40 or so for .223/5.56.
I did an Appleseed shoot last year. By far, the most popular rifle there was the Ruger 10/22. I don't think anyone had an AR-15, not even in .22 LR.
Begin at the beginning. Get a traditional rifle like a 10/22, with open sights, learn to shoot well with that, and you'll be able to transfer those skills to anything. And you'll be able to shoot more -- a lot more -- for the same money compared to .223/5.56.