The “hunting game” encompasses a pretty large arena. What game and what areas of the country did you have in mind?
The only animal I would kill is the coyote but I will keep deer in mind just in case I need meat.
Deer, coyote and long range targets - I would gear up with a long range deer rifle then - 6.5 CM or 6.5/284. Any of the standard rifles will be good - Rem 700, Win 70, X bolt, Tikka or Ruger american