Per this sub-forum's focus, that is over 600 yards for a CF. There are a lot of cartridges and I could type for hours on all of them. Typically, we are looking at .24 Caliber to .30 caliber. Cost, recoil, reload, distance, paper, steel or game animals all factor in.
Traditionally, we had .2432Win, .260Rem, 6.5x55 Sweede, a few 7s and .308Win, mostly based on .308 (short) length cases for long range. ELR was the magnums and 1,000 yards plus. Today, there are two groups "Creedmoor" (CM short Action) and "Precision Rifle Cartridge" (PRC Long Action) that are dominant. The ARC will probably be popular soon as well.
For fun, low recoil and cost, you are going to want a 6.5ish cartridge. Can be done with .22 and 6mms, but shorter barrel life, or 7mm to .30 caliber but more cost and recoil due to bullet weight. I have a few in each segment, and what I grab more than anything else is a 6.5mm.
In 6.5mm, you can go with a .260Rem, .260AI, 6.5x55 Sweede, 6.5 Creedmore, 6.5 PRC or on of the more specialty rounds like 6.5x47. 10 years ago, most of the top PRS shooters shot 6.5/7mm or .308. Now most shoot one of about 6 or 7 different 6mms. Barrels last on the order of 1500 to 2000 rounds and they spin up a new one.