Embarrassed to admit this.... but ...

[Arch]

If you have a hub gear, you can't see any of this in action though, I've yet to figure out how those work despite having one on my bike... one step at a time

Magic. That's all you need to know...

A good hub gear will just go on and on, with little maintenance, so you just don't need to know how they work. They just do. Although if you really want to know, there are different sized cogs inside where the axle turns, and when you shift gear, you change which cogs engage with each other on a similar principle of different sized cogs making the wheel turn more or less per pedal revolution. But actually, it might as well be fairies.

 

[Andy in Sig]

A rough rule of thumb ride to using gears, assuming that you have three big 'ns at the front and 9 little 'ns at the back:

When you're on the right hand big one at the front only use the right hand 3 or 4 at the back. Middle at the front, middle 4 or so at the back, left hand (smallest) at the front, left hand 3 or 4 at the back.

This is because loads of the 27 gears overlap and because you don't want the chain running at a serious diagonal because that makes it all very inefficient. If you're redundancy payment was a whopper then get a bike equipped with a Rohloff hub gear. It has 14 true gears about the same range as all the 27 but there is only one ring at the front and one at the back. And it's maintenance-free.