I think some pictures are in order here rather than have everyone try to explain.
This is a 28qt igloo cooler that I mash in. (That is letting hot water & grain sit there for an hour.) This was in my garage - you might have something similar - if not, you can buy one for $15 max.
This is my copper manifold that sits in the bottom of the cooler. The slits cut in the bottom let me pull water through the grain without letting the grain through. I made it when copper was cheaper, total cost for me was $20. Since copper costs a lot more now, it might cost $35 now. You can make the exact same thing out of CPVC pipe with some slits cut in the bottom, or with a bunch of small holes drilled in the bottom. Some people use braided stainless steel hose to do the same thing - both are cheap & easy & effective. Notice that it has a spout that comes up to the top of the cooler - this way I didn't have to drill holes through my cooler for valves. Easier was better for me.
This is the hose I attach to the manifold - I put a brass valve in the line to control how fast the sweet wort can go through the line. It is attached to the line with barb fittings. Cost: $10
These are my kettles. The one on the right is the turkey fryer - it is 30qts and has barely enough room in it to do a full boil. The one on the left is my old stainless brew kettle from when I did extract batches (20qt). Turkey fryer (pot & burner & stand) $50.
That's all you need - you probably already have a kettle, maybe two. You probably have a cooler already.
1. I heat water for the mash in my turkey fryer. (normally 1qt water to 1 lb of grain)
2. I add this water to the cooler (with the manifold on the bottom under the water).
3. I mix my crushed grains into that water & let it sit for an hour (in most cases).
4. During that hour I heat up my sparge water in my old 20qt brew kettle. (about 5 gallons to 170F normally)
5. I attach my tubing to the spout in the cooler.
6. I start a siphon on the tubing & slowly drain some of the water from the cooler - this gets collected in a 1qt measuring cup.
7. I pour that quart back onto the grain nicely and slowly (you can put a piece of tin foil on top of the grain so you don't disturb it much).
8. I recirculate 1qt at a time 3-4 times.
9. When the wort coming out of the cooler isn't clouded up with crap anymore I start siphoning into my turkey fryer.
10. As the water level in the cooler drops down I add my hot water from the 20qt kettle onto the tin foil in the grain. (try to keep 1-2 inches of water above the grain level.)
11. I continue adding the sparge water (the hot water you are adding to the grain) while draining until I have collected 6.5 gallons in the turkey fryer.
12. Boil the wort I collected in the turkey fryer just like extract.
The parts included in the bucket setup you referenced at the beginning cost $60 - for the equivalent parts my setup was cheaper - and it is also more effective because I can control the temperatures better.
Each of the steps I listed above have special names, and there is a lot of learning you can do about each step. For now, if you want to go all grain, learn as you go. You won't be able to get a good grasp of everything without doing it & learning from mistakes. Right now you should mash 1qt water to 1 lb grain for 1 hour at 150 degrees F. Heat your sparge water to 170, and try to collect 1 gallon of runoff every 5-10 minutes. That will make good beer. Specialize as you learn.
and a chance to burn money on cool hardware.
