What weight of oil goes into a push lawn mower?

That depends. Is a manual push mower with no motor, or a motorized one? Oil is usually rated by viscosity and I would use a single viscosity oil in a mower. Multiviscosity oils have modifiers to keep them fluid under cold conditions (in a car engine) but still thick when hot. In a mower a 30w oil would be ideal. Use the cheapest oil (in the long run): Amsoil synthetic.