The 6.5-300 is a pretty big step up from the .257 Roberts. Its a fast cartridge for sure. It burns a lot of powder though (twice as much as the .257), which means more expense to reload, more muzzle blast, and more recoil. And if you shoot a lot, barrel life will come into play. If you decide to go with this cartridge I'd lean toward the heavier bullets to take advantage of all that powder. 127s would be good but also check out the 143 Hornadys, 140 Bergers, 140 Accubonds, or 142 ABLRs.
The .264" caliber (6.5mm) is better in my opinion than the .257 because of the much wider selection of bullets available. Other cartridges I'd consider in this caliber would be .260 Rem, 6.5-284, 6.5-06, 264 Win Mag, or 26 Nosler. If a custom rifle is a possibility and you don't mind custom dies as well...you have even more options: 260 AI, 6.5-06 AI or 6.5 Sherman (both "improved" versions of the regular 6.5-06), 6.5 WSM, 6.5 SS.
Personally I'd do a 6.5-284, 6.5-06, or 6.5-06 improved. To me those have about the right amount of power for a .264" bullet. They'll push the 140s at 2900-3100 fps which is great for hunting most big game within reasonable distances. The larger cartridges would be better for really long range but I'd rather have a .30 or .338 for the longer pokes anyway.