Good choice of both round and rifle.
for the record, the .308 is more accurate when using 147 grain match bullets in a rifle with match barrel and receiver. I got this info from an ex-military sniper. It's also very limited in the range of bullet weights you can get commercially.
the 30-06 on the other hand is commercially loaded from 55 - 220 grains, so is able to be used to good effect for everything found in N. America from gophers to elk. (yes, Remington makes a 55 grain load - shoots a .223 bullet with discarding sabot at a frightening velocity)
that pump remmy is also very nice - good choice and good hunting!