It depends on the distance and how familiar you are with guns. If it were your first gun then I would suggest a .22LR as they're inexpensive and allow you to get down the basics of shooting. If you're experienced and need more range the Remington 700 is a great rifle. I prefer 243 for shorter distances and 7mm for longer distances. But some might suggest a 308 or 30-06 because they do have a lot of reloading options. Another goo rifle to check out would be the Ruger HM77.
Pick a good .22lr bolt rifle Savage or CZ then go to walmart and bu a low cost centerpoint 4-14 scope and buy some better 1050fps target/match ammo and start off at 25 yards, work out 100 yards. Excect to try atleast several ammo brands before you find one your rifle likes well enought to enjoy. Don't expect bulk boxed ammo to shot well. Most clubs have .22lr matchs too.
After you master all that can with the .22 move up to a savage F/TR in 223 or LLR in 260 or 6.5 creedmore and shoot out to 500 to 600 yards.
+1 on the 6.5 family (6.5x 284 1st choice, 6.5 Creedmoor 2nd) I prefer Sako because they guarantee MOA and are $600 on sale. If they don't guarantee their work, it could be great, average or so-so, ya just never know until AFTER you buy it.