Should I take Ohm's law for granted?
Yes, you should take Ohm's law seriously.
You do, though, need to keep in mind that it applies only to simple resistors and conductors.
Ohm's law is a simplification of some complicated math.  It applies only to linear resistive circuits.  If you stay in that area, then Ohm's law will give you correct results.
The other elements you mention (LEDs, transistors, diodes) are not linear resistive elements.  Those parts have a very different relationship between current and voltage.
For diodes, you can refer to the Schockley diode equation for the relationship between current and voltage.  It also applies to LEDs, which are light emitting diodes.

The simplification given by Ohm's law is often times all you need.
Take an LED as an example.  A typical task is to calculate the value a series resistor in order to safely operate an LED from a given voltage.
If you just connect an LED to a voltage source, you'll destroy the LED.
What you do then, is to look up the maximum safe operating current for the LED as well as its nominal forward voltage (both given in the LED datasheet) then use Ohm's law to calculate a minimum resistance.
Say you want to operate a blue LED from a 5V source.  You look in the datasheet of the LED and find that it can tolerate a maximum of 20mA and that it has a nominal forward voltage of 3.3V.
That is to say, when given 20 mA a voltage of about 3.3V will appear across the LED.
The difference between 3.3V and 5V is 1.7V.  At 20 mA, Ohm's law says you need an 85 ohm resistor in series with the LED.
That won't be perfectly correct, but close enough.  You will then usually find that you need a resistor with a larger value because that 20mA maximum is really bright with modern LEDs.
If you actually measure the voltage across the LED or the current through it, then you will find differences to the calculated values.  The voltage drop across the resistor will obey Ohm's law, though.
Keep in mind that a lot of what you learn in the beginning when studying pretty much any subject will be simplifications.
If you were to start studying electronics, and your instructor pointed you at Maxwell's equations and told you to calculate the current through a resistor for a given voltage, you'd probably just quietly leave the classroom and never come back.
Ohm's law itself is empirical, and only applies to purely resistive circuits.  Purely resistive circuits don't exist - every conductor and every circuit has inductive and capacitive effects, as well as depending to some extent on temperature.
Learn Ohm's law, use it.  It gives usable results over many common conditions with many common materials.  Just keep in mind that is doesn't cover all conditions or materials.