Minecraft – How to get this setup to work properly

If you watch at about 2:50, you can see most of the left half of the circuit. The part for the stairs looks like it isn't shown very well but that is less important timing wise. Watch carefully over and over and see if you can replicate it this way. It looks like the repeaters for the middle activate and deactivate immediately. So the order (from a circuit standpoint) for both directions is M, L+R. This works for closing because the middle repeaters don't have anything to activate until the blocks are pushed into the middle. When opening, they will retract the middle immediately.

Breaking it down:

Opening

1. Middle repeaters deactivate (middle pistons retract)
2. Delay
3. Left and Right redstones deactivate (left and right pistons retract)

Closing

1. Middle repeaters activate (nothing happens because the middle pistons are not in position)
2. Delay
3. Left and Right redstones activate (left and right pistons extend)
4. Now the middle pistons have been pushed into position and are activated by the middle repeaters

You can start making basic "redstone" devices as soon as you have enough wood to make a door and a pressure plate: make the door and the pressure plate, and put them next to each other. Voila! Your first "redstone" device without needing any redstone.

If you want to start getting into more complex devices like flip flops, you need redstone dust: you can find it by mining redstone ore deep underground: 16 layers above bedrock or lower (sea level is at the 64th layer).

The best way to learn how to make redstone devices like flip-flops is to watch the myriad of tutorials on YouTube: figure out what you want to make, search for a tutorial on how to make it, and repeat it¹.

For example, Rolf-David's tutorial shows you how to build a basic T Flip Flop:

The basic flip flops (and most redstone devices) are pretty simple, but aren't very efficient because of the way redstone works: they take up a lot of space and because power doesn't travel instantly across redstone, larger circuits are incredibly slow.

So what people have been doing is exploiting buggy implementations of blocks not intended for redstone circuitry to make them smaller and faster. Etho is probably best known for exploiting what's called the "block update"; a quirk of Minecraft where blocks send power when they update their status:

For the most part, if you never got into advanced features like this, you'd still be fine.

^{Note 1}If you want a more comprehensive education on Redstone, there are couple of channels that focus on it pretty heavily: Etho's Lab and SethBling probably the two biggest ones. Etho's Lab is more "Let's Play"-ish (although he does do stand-alone tutorials from time to time), whereas SethBling is pretty heavily focused on tutorials.