For the game you mention I would choose the 243 Winchester. If you prefer over kill then you can bump her up as much as you want.

I really think the 243 Winchester would be a great first center fire rifle. And for a first rifle I would encourage one to go with a rimfire, probably a 22 long rifle. A bolt action would be hard to beat in either the center fire or the Rimfire rifle.

As for a choice on what gun I would encourage one to give the cz452 22 rimfire a close look. If you can come across a good used Winchester model 70 in 243 I expectyou would be happy with those.