Using a transistor correctly to "connect" Maximite computer and circuit
I looked up the datasheet for an IRF530 and see that the "Gate
  Threshold Voltage" is between 2.0V and 4.0V. If I understand things,
  that means that this MOSFET will "turn on" when the gate is given
  between 2.0V and 4.0M. Is that right?


No, it will begin to turn on at a voltage that may be as high as 4 volts. By turn on, it guarantees that with a gate voltage that might be as high as 4 volts, the drain current is 0.25 mA.

This characterisitic is better suited for understanding how it might turn on properly: -



If the gate source voltage is (say) 5 volts then typically you could expect a drain current of 1 amp would drop about 0.2 volts across the device.

Regarding the circuit in your question, you have drain and source the wrong way round and you will need a pull-down resistor (say 10 kohm) between gate and source. Here's a picture from the internet that gives you the general idea: -