In preference I would like to use a breadboard to work on but doubt it can hold the power.
Quite correct, that motor would melt a breadboard.
As to the schematic you need a 10K pull down resistor from the gate to ground on the FET, and I would put a 150R resistor in series with the gate.
However the FET is totally unsuitable, it requires a 10V gate signal to turn it on and the rated current of 35A in a TO220 package is too close to what you want to drive. FET headline currents can rarely be achieved in practice due to thermal considerations.
I would also wire the push button properly, that is with the buttons between input and ground and an external pull up resistor of 1K. With that sort of load using the internal pull up resistor would not have enough immunity against interference.