You’re pushing the atoms on your end, which in turn push the next atoms, which push the next ones and so on up to the atoms at the end of the rod which push the hand of your friend on the moon.

As it so happens the way the atoms push each other is electromagnetism, in other words sending photons (same thing light is made of) to each other but these photons are not at visible wavelengths so you don’t see them as light.

So pushing the rod is just sending a wave down the rod of atoms pushing each other with the gaps between atoms being bridged using photons, so it will never be faster than the speed at which photons can travel in vacuum (it’s actually slower because part of the movement of that wave is not the lightspeed-travelling photons bridging the gaps between atoms but the actual atoms moving and atoms have mass so they cannot travel as fast as the speed of light).

In normal day to day life the rods are far too short for us to notice the delay between the pushing the rod on one end and the rod pushing something on the other end.