The minimum caliber I would recommend for the various kinds of game you want to hunt would be .260 Remington, 6.5X55, or 6.5X57. Any of those calibers would do the job and the deciding factor would be the availability of ammunition and the cost. The .260 is simply a .243 with a larger diameter and heavier bullet and the performance is virtually identical to the other two mentioned.

S&B lists only one bullet weight for the 6.5X57, but three for the 6.5X55. However, if you take one further step, there are four bullet weights to choose from in caliber 7X57, which would be my recommendation.