For target shooting at around 200 yards I would suggest the .223, cheap ammo, everywhere available, low power, accurate.
If you like a caliber that takes a punch take the .308, its good for shooting out to 500-700 yards and it is in the mid-range power-wise. Its the caliber of choice in most countrys as ''sniper'' cartridge in the military like NATO. Its not so cheap like the .223 but also not very expensive. You can find this ammo everywhere.
If you want a very famous caliber that is a tad more powerful than the .308 than take the .30-06. It shoots slightly heavier projectiles as fast as the standart .308. Dont even ask about ammo availability, its also everyhwere! There is .30-06 ammo that is cheaper than .308. but also has poor quality. So the ammo price is on par with .308.
Nearly all important rifles are chambered in that three calibers.
I personally would go with the .308, its THE perfect all-around caliber today.
But as always: First rent a rifle in the caliber of you choice and shoot it. Then, buy!
Actually this is one of the reasons I have always opted for the .308. It’s a NATO round and as such it has the support, backing, need, whatever you want to call it of several governments, ours included, thus insuring its continued availability.
This is another big reason to get a .308.