Your question is two parts: Which rifle and which caliber (.308 vs. 30/06)
Caliber first: I have taken a number of elk since I was sixteen and I have used the .243 Win., .257 Roberts, 25/06 Rem., 280 Rem., 7mm Rem. Magnum, 30/06 Spgfld., .308 Win., .338 Win. Magnum and 45/70.
Two things they all have in common is they are still dead...and they were all shot at less than 200 yards.
In-my-opinion the best performing calibers were the 30/06, .338 and 45/70...and what they all had in common is I used bullets weighing 180, 220 and 365 grains, respectively. Over time I have became a proponent of heavy bullets at moderate velocities.
Since you are asking our opinion on a choice for a primary hunting rifle, in my case it would be a synthetic stocked Kimber, Winchester Model 70 or Remington Mountain Rifle with a 2.5x-8x variable Leupold....in 30/06!!