I too, vote the 308 Winchester as THE supreme "medium sized gun" caliber. It's the cheapest big game suitable cartridge to reload for (burns less powder, and 30 cal bullets have some very cheap options compared to anything else but 22 cal), the cheapest to buy ammo for, it's "enough gun" for anything on this continent out to at least three times the range a newbie has any business flinging lead at a critter, it doesn't kick very hard, any store that sells ammo will have a box of 308 somewhere (take that, 7mmWSM fans!), etc, etc, etc.
There are all sorts of calibers which are wonderful, and very suitable to what you want (7-08, 30-06, 270, 280, 6.5x5, 260, etc, etc, etc, etc, etc, etc, etc.), and none of them are in the least ways poor choices. But, a 308 is a very very good choice, and for the reasons stated - I think it's the best.