Well audio preamps and power amps amount to sequential gain stages and you can see it on an ocilloscope.
You would probe the audio signal at different stages in the chain or circuit and see the amplitude of something like a sine wave. You turn the knobs on the scope until you get a nice peak-to-peak signal visible then read the voltage swing from the scale. Digital scopes do some of this for you and you can play it back.
Your signal is really a type of A/C voltage, being amplified by those transistors which are being employed as gain stages.
Now a preamp may only put out a few volts peak-to-peak but it can start with a minuscule millivolt source so that's a lot of clean low-noise gain that's desirable. It's just an amplitude enhancer, low signal voltage in, higher signal voltage out. These are the small low-current usually plastic transistors that are not too expensive and rarely if ever need a heatsink. They sure can be made to sound good though.
But they can not pass the kind of wattage and resulting heat through their little silicon crystals for you to get real movement from any sizable speaker and that's why the preamp ends there. Headphones can be good sometimes though :)
Preamps don't really have output measured in watts, they mainly add gain to the amplitude of the signal.
And the same amplitude can well drive a choice of power amps to whatever wattage you need depending on the power rating of the amplifier.
Power amps are current amplifiers mainly, with heavier components like transformers and metal-encased transistors containing much bigger chunks of silicon, bolted on to heatsinks often with fans. With fatter copper wiring too so they can move the higher amounts of power.
But both pre- and power- amps will have increasing amplitude on the scope as you crank it up.
Now imagine an amp powered by a 12V car battery which is ideal for a preamp. That 12VDC is your "headroom". The maximum sine wave you will be able to get out of it will be 12V peak-to-peak. Depending on the signal coming in and how the gear is, if you (can) give it more gain beyond that then the tips of the peaks will be "clipped" on the scope where they appear with flattened tops & bottoms. The volume will still be increasing as you turn it up more but you will hear the distortion more prominently as it gets further clipped. This is the onset of "overdrive" where you can be putting a clean input signal into a decent gain stage but if the gain is too high you're asking too much and you no longer get the fully proportional "linear" response from the output which is what high-fidelity is supposed to be based on.
Keep cranking it beyond that point and you can get amplitude so high that the sine waves are so far off the top & bottom of the scale that they are functionally square waves. This is actual signal "saturation" where all kinds of harmonics are mathematically there which were not present in the source material. On music this occurs in a dynamic way.
The analog VU meters used in vintage studios were logarithmic and could be set to zero with a scope and a test signal, beyond that was in the red. Segmented LED indicators can be calibrated the same way, the response in action is just more instantaneous.
If you're going to watch the output of some random things you're going to need a 'scope. The music does look mesmerizing.
There are some scope apps for smartphones or PCs but if it's too much spl the mic can be the cause of erroneous readings. To really visualize electrical response in real-time you would need a USB interface with actual probes and connect them to equipment outputs or circuit test points.