If amps and volts can vary independently, how does Ohm's law actually work?
The short answer is that, if you are varying I and V independently, you must have some means of varying R.

For a fixed R, you understand fine : as you increase V, I will increase dependent on V, at a rate which is exactly the resistance R. Lock in that understanding, it is important.

But that's a special case. 

You often want to vary the power level from a fixed voltage - that means varying the current, which generally means varying the resistance. And there are various ways of doing that.

A crude way is to vary the number of loads on a circuit- switch on additional lamps on a lighting circuit, or heating elements in a heater.

Another way is to switch the resistor on or off relatively fast, so its resistance alternates between R, and infinity, to reduce the average current. Examples are thermostatically controlled heaters (or the simmerstat controls in ovens) and PWM "pulse width modulation" used for dimming lights and some motor speed controls. These can only reduce power by increasing the effective "R" - once R is connected all the time ("100% duty cycle") you are at full power.

(Motors are a bit more complicated than that. because motors are also generators. Motor control will keep for another day)

There are also resistances which are variable in themselves - either as a function of temperature (which is a function of the power dissipated in them) or as a function of the voltage across them, or some other effect (like light falling on them). Ohm's Law still applies ( V = I * R ) but R is no longer a constant, and the equation may not be a linear one. 

Some of these devices are semiconductors; but consider a simple incandescent lamp bulb first. As its filament wire heats up, its resistance increases, and is more than 10x as high at full power than it is when cold.