For the stepper motors you need 4-core CY cable, which is shielded. If the cables are quite short then 1mm^2 is fine, but for a big machine I'd use 1.5mm^2 to reduce the voltage dropped across the cables.

CY Cable - 4 Core 1.0mm2

For the signal wires in the control box any thin wire would do and shielding whilst not necessary might prevent problems.

You need fairly thick cable for the power to the drivers - similar thickness to the motor wires, so you could just use that if it saves buying more cable. Use separate wires to each driver - don't connect the power from one driver to the next to the next...

A contactor is a switch that is controlled by an electric current, not by pressing it. Rather like a relay.