edit: I misremembered the size of the hole saw, I used 32mm not 1/2"
It's a very basic ebb/flow system. There's a tank of water at the bottom, a tray of plants on top, and occasionally the water gets pumped up to the top to feed the plants.
MATERIALS:
Tubs [~$15] I used sterilite containers like this. The top one should be about half the height of the bottom. Using clear container will make your life easier.
Fittings [~$11] Amazon or get some somewhere else. I don't think there's much difference.
Pump [~$15] Amazon make sure you get a submersible pump so you don't have to worry about an intake line. Find a guide to help you calculate the correct GPH for the size tanks you get, or it either won't fill properly or will make a spout.
Tube [~$0] My pump came with a tube. If yours doesn't just make sure to get the right size for your pump and fittings. The fill fitting probably has a smaller size than the drain so make sure you don't get mixed up.
Timer [~$15] Amazon This is what most people use to control the flood cycles. If you have some kind of smart home system you can use an outlet to control it remotely instead.
TOOLS:
Drill [~$?] My 12V cordless cut through the tubs easily enough, I don't think you'll have any trouble here. Use whatever you can get your hands on.
32mm Hole Saw [~$10] Amazon these worked ok, but scuffed up the surface around the hole a lot. If you care about how it looks maybe find a better way to make holes. Also make sure you match the hole size to whatever your fittings need. In any case you probably need at least 32mm to fit the pump cord through.
ASSEMBLY:
Use the hole saw to drill 2 holes in the top of the smaller bin. Add/remove adjusters to get the fill drain to the height you want. Install in the holes.
Drill a hole in the side of the large tub. Place it as high as you can without it overlapping with the small tub. Place the pump in the tub and route the cord through the hole. Attach the tubing from the pump to the fill fitting.
Fill the large tub with some water, at least a couple inches more than your desired fill height for the tray. Place the smaller tub on top. Plug the pump into an outlet. The tray should now be filling. Once the water starts flowing through the drain fitting, unplug the pump. Make sure that the water drains properly. I found it didn't drain very well so I drilled a small (~1/8") hole in the middle of the tray. It's not enough to prevent it from filling, but now it drains very well.
Configure your timer to whatever flood cycles you want, connect that to an outlet, then plug your pump into it.
GROWING:
As for grow media, nutrients, pH levels, flood cycles, you'll have to find that elsewhere. I don't know what the fuck I'm doing.