A thirteen year old boy should be able to handle the recoil of any of the standard cartridges with out any problems. I got started with a 12 gague at eleven and a 30-06 at thirteen then traded it for a 7mm mag when I was 14. Unless you are particularly recoil sensative then the regular old .270 Winchester would be about the ideal catrtridge for you. It is very flat shooting with the 130 grain loadings and recoil is pretty light. It also has the ability to use heavier 150 grain bullets for elk and moose later on if needed and recoil is still on the light side. Bullets are available at every store that carries ammo for anything so you will never be with out ammo and it is inexpensive compared to many of the other cartridges that are not as widely available. Every rifle manufacturer that I know of chambers for the .270 so you should have no problem at all selecting a gun that you like and is within your (or your parents) budget.

My ten year old son loves to shoot mine every time we go out to the range. He has no problem with the recoil at all. If he can shoot it then you should have no problem at all. Mine doesn't even have a recoil pad, just a plastic but plate. With a good recoil pad the recoil feels even lighter.

Good luck.