The 7mm-08 I think would be just fine. If you want load info or build info let me know. Anything .260 and up is capable of knocking a deer down at 500+ meters. The question is are YOU???
It's all about on game performance and accuracy. Our premium bullets of today are capable of wonderful things at many velocities and ranges.
I must stress again that for a new shooter you have to start baby steps and with mucho practice and you won't get this out of a shoulder pounding battering ram. 20 rounds from a powder eating LA or Magnum will surely not encourage you to wanna practice.
The 6.5mm ,7mm , 7.62mm bullets in SA chamberings will do everything you want and more. You can blast out a couple hundred rounds in one sitting and not be bothered by recoil.
Look at what most long range competitors shoot, that will answer a few questions.Again anything using a .308 case is a good choice as components are available and affordable.
My hunting round for 7mm-08 stays supersonic out beyond 1300 meters and the velocity/energy for proper bullet expansion in the 800-900 meter range to drop deer sized game, and shoots 1/2 MOA
Again, you gotta be GOOD to drop game at those ranges, practice and practice and practice.