Everybody has his individual approach to electronics. I started by measuring the characteristic curves of a resistor, diode (forward and reverse), Z-diodes, LEDs, and finally of transistors. The transistor circuit required to also understand Ohm's law, at least after killing the first transistor. After drawing some characteristic curves myself, I learned how to interpret the diagrams in the data sheets. Then I continued by building up simple circuits, then modified some parts and measured the difference to the original circuit design.
You may read about Ohm's law first, and how voltages and currents distribute across resistors in parallel and in series. Then verify your calculations by experiments.
Data sheets are confusing by a sea of numbers. Some experience is required to find the essential numbers, which for simple items (except transistors) boil down to the maximum allowed voltage, current and power. With transistors also the "behavioural" constants are important, e.g. the required gate voltage for turning a FET fully on and off.
That's enough for a head start, later on some understanding of capacitors will also be required. Then you'll understand why a FET gate should not be connected directly to a digital output, instead a gate resistor should be inserted, which limits the current while the gate capacity is charged and discharged. For safe discharge also a resistor to Gnd (gate-source) should be added, so that the transistor is turned off even if the microcontroller pin is reset to an input and cannot actively pull the gate low.
Motors. and other parts containing coils, also show some strange behaviour. Current will increase slowly when a voltage is applied, but if you then disconnect the cable by hand, you may experience an electric shock - the EMF strikes back! That's why seemingly useless diodes or other snubber circuits must be added, which allow the back EMF current to find a safe way, around the sensitive hand or switching transistors.
It also may be astonishing that electronic parts are not ideal, they always have some tolerances, and many parts are dimensioned by rules-of-thumb instead of exact calculations. But I'd suggest not to blindly trust such rules of other people, instead verify the results by calculations and experiments, until you know which rules are right and useful. As you may have read already, the parts and circuits in the Starter Kit are not always well chosen and designed. In the long term it's not sufficient that some circuit happens to work for somebody, instead it should work for everybody, under all reasonable circumstances.
Electronics artwork is based on experience and a few basic circuit patterns, followed by a numeric verification of the chosen operating point(s) and resulting limits, and by final tests of the life circuit.
BTW: the mentioned IRF540 data sheet roughly applies also to your (and my) IRF520. Compare the mentioned values yourself, and try to verify the more precise parameters of your FET in a test circuit. If you don't know how to build such a circuit, simply ask again 
If I understood correctly that means all of these fets would need 20V to fully open are are incompatible with the Arduino?