Welcome to the forum. You've got a nice rifle to start with. There's plenty you can do to improve the accuracy potential. Here's a couple examples of things you can do....
My 308 1k Rifle - The Evolution Continues!
Nephew's Transformed Rifle!
In terms of ammo to shoot, always consider the bullet first. It's the part that is ultimately going to get the job done. Generally speaking, if you can get better than 1moa accuracy out of your rifle, you should be able to take deer at 500 yards.
If you're willing to spend $27 a box for ammo, give serious consideration to saving a few $$ and start reloading. Then you can choose the right bullet for what ever job you have.