Good suggestions by all. It boils down to what you like and what will be the largest game taken. My choices would be 7-08, 260 Rem or .25-06 - in that order. The .260 Rem is pretty much the equivalent of the 6.5x55 but fits a short action very handily and ammo is more available.
Do you handload? If so, another good choice is a .308. In fact, with the new lower recoil rounds, it's a good choice even if you don't handload.
The Ruger Compact, along with the Model 7 or youth models of various brands, would be a good choice. The fact that you can replace the stock later with a full-size one as the child grows is an added benefit.