Welcome to LRH!!
The Rem 700 is a fine accuracy rifle. I would have to honestly say that as a whole the older rifles are built a little better.
I have just seen a few specific rifles come through my shop with some quality control problems that certainly should have never gotten out of the plant at Rem.
Still on average, even the new Rems are top performers in the accuracy department. As a gunsmith I would say build a rifle but this is not always practical and I understand that.
For your money, you have a better chance getting a fine shooting Rem then many other brands for the money.
It may also suprise you to hear that the Savage rilfes are generally very good shooters. They are not flashy or fancy but I would say for the money they are the best shooters on the market. In the same class as the Rems but for a bit less money.
THe 308 adn 30-06 are great hunting rounds for what you are planning on hunting. There are also great bullet selection for long range shooting.
It all depends on what distance you plan on shooting and if you will just be paper punching. For pure accuracy the 308 is a great long range round.
For long range hunting past 600 yards or so, I would highly recommend going up to a 300 WSM
class. Like I said, just depends on your range goals.