The Remington 700 5R is easily one of the best values in an out of the box precision rifle. They use the HS Precision made stock with a bedding block and the same style 5R barrel that the Army uses on the M-24. They are made in 223, 308, & 300 Win Mag. They have a well deserved reputation for outstanding accuracy and I have seen several of them shoot 1/2 MOA with good factory ammo and better than that with handloads. They can usually be found on Gunbroker for about $1000.
On caliber, there really is no better round to learn LR shooting with than the 308. It is accurate and efficient with moderate recoil.