For a mid range sniping (700-800 meters). The 7.62x51 or 7.62x54R are good.
For an assault rifle you want something lighter. Controlling a 7.62X51 NATO on full auto is not very easy. you need something lighter. In that range you have: 6.5x55 (Avtomat Fedorova, back in the beginning of 20th Century), 7.62x39 (bad ballistics passt 300m), 5.56 NATO, 5.45 Russian (those two cartriges are too light IMHO and lack AP capability), 243 Remington, 6.5 Grendel (6.5x39), 6.8SPC. I would say that for an assault rifle, for accurate fire for 500-600 meters you want either those three or 6.5x55. A bullet of about 120-130 grains, with moderate caliber accelerated to about 820+m/sec