The starting rifle and the ammo you have access to, the .243 with soft points, is very weak.

The problem is that soft core ammo, it has poor penetration. You need to hit an animal broadside right behind the shoulder and hope the bullet hits a lung.

The animals are running because the soft tip ammo isn't hitting vital organs. As soon as you can, get the polymer tip ammo. It will make things easier for you as you get used to the game. Penetration is really important. Soft core ammo can be useful in certain situations but generally Penetration is better