Well, I'll start this one out! What is the intended purpose of the rifle? Punching paper (if so, just casually vs. you want to get into Benchrest or something else specific), clanging steel, hunting game, varmint control, self defense, etc.? A combination of a few of these? Most rifles over .22LR will be more accurate than the shooter from 0 to 300 yds. What's your definition of accuracy (1/4" groups at 100 yds, 1.5" groups at 100 yds., etc.)?
Have you shot other rifles in the past, or is this one to learn on?
Do you "like" or have an interest in a certain type of rifle? Bolt action, semi-auto, lever action, etc.?
What's you price range - $300 Savage .223 or $3000 GA Precision 6mm Creedmoor.
Do you have a need for optics?
Do you reload or have an interest in getting into reloading? Makes some of the more expensive cartridges much more accessible.
Not knowing any of this, I'll begin with a .223 from one of the major manufacturers will get you to 300 yds. easily with great accuracy, low recoil and low cost. Something more off beat/unique might be a 6.5 Grendel for instance, which will do 0 - 300 yds also and make larger holes in paper which might be easier to spot for those of us with older eyes. Give us a bit more info. and folks will be able to narrow down their suggestions. Cheers!