How to watch inductor and capacitor

You'd need more sophisticated test gear to do that properly, typically a two (or more) channel oscilloscope with a function generator (to produce the sine signal). 

There's no need to know this visually (although it helps) as the mathematics takes care of this for you using the magic "imaginary" number to represent the weirdness.

Actually, it just struck me (and I should have thought of this earlier but I've been kinda busy) but you can use this circuit simulator by Paul Falstad:

https://www.falstad.com/circuit/

The example that's currently on the home page actually shows an animation using a resistor, inductor and capacitor. You can see how the electrons (the current) actually move through the circuit.

It's interesting to note that nothing actually passes through capacitors it just swaps sides. Many textbooks seem to gloss over this for simplicity but it's useful to know.

You'd need more sophisticated test gear to do that properly,

Indeed, but this is where modern graphical simulation can help, don't you agree? The sort of gear required to do this IRL is often beyond most weekend hobbyists unless they have more money than sense... I'm reasonably advanced now but my gear is "modest" not to put too fine a point on it. 🙂 So I rely on simulation to check my working and try to design to match the theory as well as possible. But yeah, this is a difficult concept to grasp mathematically and even harder to see.