Are you using an UNO? You can use Serial.print(ln) instructions. If the Arduino is the DTE device, you may need to set RTS and/or DTR. If the Arduino is the DCE device, you may have to set CTS and/or DSR.
Look for: Logic-Level Gate Drive Low value for RDS(on) (Ω), IMO I like less than .1 Ohms(100 milli ohms) (lowest possible) Drain-Source Voltage, to be greater than your power supply Continuous Drain Current, to be greater than your load will require
I would switch the HOT wire (smaller of the two plug holes) not the Neutral wire. I would also connect the earth ground wire to the motor frame if it is recommended to do it. You may find you need a contractor too switch that much current.
That's the point: what if i connect both vin and power jack?
No. You can use the power jack and USB at the same time. If you wanted to use Vin and the power jack at the same time you need to add a diode between Vin and your external power supply. Cathode goes to the Vin pin. 1N4007.
I cannot read what is connected to L or R. IMO, I would use GND wire. What kind of bike? I normally use 220R for most LEDs at 5v power supply, but, that depends on the colour and brightness of each led. 1N4004 is ok.