Calculating the resistance?

How do you calculate the resistance for this type of motor driver circuit?

I'm using an IRF520 NMOSFET and the pump (motor) voltage is 12V DC and current is 300mA

Which is not a logic level mosfet, and basically a poor choice.
But maybe ok with this light load (if the fet isn't getting hot).
You can find the 'on' resistance with 5volt at the gate in the "IRF520 datasheet".
This mosfet is unsuitable for 3.3volt processors.

You can't, because transistor characteristics are given only approximately in the data sheet. To measure the channel resistance, measure the drain to source voltage and divide by the measured motor current.

To emphasize the previous post, this is the wrong choice of transistor for Arduino circuits. You want a logic level MOSFET.

See this thread:

For this circuit the possible range of resistances and the lack of critically make the resistor values more intuitive than calculable.

R1: Unless you are trying to PWM the motor at frequencies exceeding 1000 HZ R1 can be almost anything from 0 to 100k.
You don't want to use 0 because a short in the MosFet will damage the Arduino. So 180 is fine and 1k is fine. You wouldn't want to go much about 1k because noise might cause the MosFet not turn on or off cleanly.

R2: This resistor is only needed to keep the MosFet OFF when the Arduino is not connected. It should be high enough no excessive load is put on the Arduino output.

Why would you want to? the winding resistance of the motor is the important thing, then you can determine the maximum current requirements, and this affects the choice of switching device and wire gauge.

Are you asking about how to choose a MOSFET?