There are a number of ways to learn, but I think videos on youtube are the best.
This tutorial won’t get into using one of those diamond edge tools where you slide the knife forward and back across the diamond surface, which is mounted in a slot. Those things are pretty good, if you don’t want to learn the freehand method. I don’t own one, but I did like using one.
Stones
I would start out with a cheap $5 stone from the dollar store, and a sturdy, but low-end, stainless-steel knife with a flat, not serrated, edge. These are the hardest to sharpen, but also the most plentiful, and cheapest.
If you have a good, high carbon steel knife, you’re lucky. You can invest in a better stone of 1000 grit or finer, and keep that knife sharp. These stones are $35 and up. I like 1000 grit because it’s a medium coarseness, and takes off very little metal. I don’t have a finer stone, and I’ll explain why later.
I’d recommend getting the cheap stone. We all have some annoying stainless knives, and you really need to be able to scrape the hard steel to develop an edge.
This dude here uses a cheap stone:
The 19 Degree Angle
You usually hold the knife against the stone at a 19 degree angle. That’s around 1/3 of a 45-degree angle, or slightly less than 1/2 of a 45-degree angle.
If you have a well defined bevel, you can copy that angle. You’ll feel it.
Pushing or Pulling
There are two motions: pushing to grind down the metal severely, and pulling to more gently grind the metal and form an edge.
If you have an extremely dull knife, you will need to use the pushing motion, where the edge of the knife pushes into the stone.
For a very dull knife, you will need to spend a lot of time grinding. Follow the video, and note that he does five strokes on one side, and five on the other side. You need to use and equal number of strokes to make sure you remove an equal amount of metal from each side.
Repeat that 10 or 20 times, and see the effect on the knife. A dull knife will start to cut, roughly.
Once the knife is cutting roughly through paper, you can change your motion to a “pulling” motion, where you drag the edge over the stone. You might also start using the medium grit side as well.
Pulling is just more gentle than pushing. It removes less metal.
What’s Happening to the Blade
The edge of a knife isn’t smooth: there are tiny microscopic “teeth” on the edge, somewhat like a saw. The coarse grit side of the stone leaves large teeth. The medium grit side knocks down the high spots and turns them into many more, smaller teeth.
These smaller teeth feel “smoother”, and are also “stronger” because they’re shorter and less prone to bend and flop over under pressure.
This video shows some chipped and rolled edges under a microscope:
Here’s a video where they play with the “burr”.
When you sharpen the knife, it redefines the edge so it’s like a tiny saw. Sharpening also produces a “burr”, which is the metal that’s coming off the knife. You need to remove that burr by grinding the edge a little more, and then “stropping” the knife.
Stropping is usually done on leather strips embedded with a polishing compound, but you can fake it. Just use a piece of corrugated cardboard, which has clays embedded in the paper.
Pull the knife across the corrugated cardboard, like you did across the medium grit stone. You can also cut into the edge of the cardboard to remove the burr and polish the edge.
This guy is stropping on cardboard:
https://www.youtube.com/watch?v=vKl7qNzbFtQ
These videos show guys using diamond hones. I haven’t used diamond, except in those easy-to-use sharpeners, but I don’t really like diamond. It’s an extremely hard material, and will grind away a lot of the metal.
I prefer the regular aluminum oxide stones. That material is also hard, but not as hard.
Finding a Better Knife
The easiest knives to sharpen are made of “high carbon steel” or “carbon steel”. They may be stainless and called “inox”, or they may stain.
Stainless steel is tougher, and also bends. Carbon steel is harder, and chips.
In a knife, you want that chipping quality, because it helps you create a better edge.
Carbon steel knives at a kitchen shop tend to be expensive, but you can find them for as little as $15 each online. They aren’t as fancy, and the metal isn’t as good, but it can be sharpened, and can perform well. It’s far better to have a knife that’s easy to sharpen than one which holds an edge for a long time.
Another way to find a good knife is to look online and buy a used knife. Most people with nice knives just don’t use them too much. They may not bother to sharpen them.
I would just look for some common, popular brands, like Sabatier four star elephant (there are multiple, different Sabatier companies), Henckels, Wusthoff, Global, Mac, Victorinox, Dexter Russell, EKCO, Forschner,
You can also find them at a thrift shop, but that’s going to be difficult, because there are fifty crap knives there for every one good knife.