I don't usually use very complex network systems and despite some time of testing, I haven't been able to get the new depot circuit network to work.

Is there a mini-tutorial?
I want to do a "simple" concept. I have two supply depots, quite separated from each other, that deliver the same material. One of them works as a backup and I want it to be used only if the first one is dangerously out of stock, either by activating the secondary depot or the belt that feeds it.

You can use the circuit writer to set an enable condition, and the road network reader to see how much is in the network

So you put down the road network reader, and set it to read the signal you want (like, iron plate),
Then put down a depot writer facing the supply depot, and make sure it connects (will show some text),
Then wire the reader to the writer, and set the writer condition to like, "iron plate < 2000"

Thank you. Following in your footsteps, I've already got it set up,

I was overcomplicating myself

Yes, but then writer activates the depot and condition on road network reader changes immediately so the supply depot will constantly be setted to be ON / OFF until "iron plate < 2000" condition false. I think you need additional condition.

Like slaved34ler said, creating a circuit this simple will make the depot to switch between enabled and disabled constantly.
You could use SR Latch to fix it. Just follow the guide on official factorio wiki that uses accumulator as an example.