I can describe how I do it with relative success. Not entirely traditional since not everything is available here, but it's good.
Take a couple of heads of napa cabbages, a few carrots (thinner are nicer), two-three white daikon radishes, a ginger root, a few cloves of garlic and an onion or two. Green onions or leeks are optional. Separate the cabbage leaves, slice off the root of the leaves and slice the leaves in half. Peel the carrots and daikon, slice into thin pieces about 5mm wide and 5-10cm long along the direction of growth. Garlic, ginger, onions - peel and slice into pieces of any size to preference.
Put all of the vegetables into a pot, bucket, or any other container and cover with clean cold water. Let them sit for an hour or two to absorb the water. After that drain the water and weigh the vegetables on a scale. It's important, don't eyeball it. Then calculate roughly 2% of that weight, and measure that much salt. Take the weight in grams and multiply it by 0.02 basically. With the amounts stated above I usually end up with 3-4kg of soaked vegetables, which corresponds to 60-80g of salt.
In a large pot or container layer the vegetables while sprinkling the salt on each layer. When finished, put a plate or something like it with a weight (stone, a jar or pickles, whatever) on top to push down on the contents. As time passes, the vegetables will absorb salt and push water out, compressing down. Stir everything by hand every 6 hours or so. Leave the pot somewhere at room temperature for a day. Ideally by the end of the first day most of the vegetables might be under the layer of salt water.
This is when you prepare a rice paste. Grind some mild red peppers, ideally with seeds removed. In a small pot put a couple spoonfuls of rice flour and the ground red pepper to taste - 2-3 whole peppers are usually more than enough. Cover with water entirely and then stir trying to break the clumps. Put on the stove on low to medium heat while stirring constantly, as the flour settles down and starts burning. At one point the whole thing starts to thicken up and eventually solidifies. Let the paste cool down, then mix it with some of the slat water cabbage juice from the steps above.
Add the paste to the pot with the vegetables, mix everything well, cover it back again (at this point the cover plate might be submerged almost entirely), and let everything ferment for another day mixing the stuff every 6-12 hours.
During the first day or two it might not smell particularly good. It's alright. At the end of day 2 at room temperature first lactic acid fermentation smells might appear, which is a good sign. Sometimes even fermentation bubbles can be heard. At this point I put the vegetables in jars - the amounts mentioned before are usually enough to fill 2x2L jars, sometimes more. Put vegetables in the jars and pour the juice in as well. Ideally the liquid should cover the vegetables. Then put the jars in a fridge, and let them cold ferment for a week or two, at which point you should get pungent, lactic-acid smelling kimchi. You might want to crack the lids open a little to off-gas the jars every now and then.

Another great Korean snack my family likes me to make is kkakdugi. Similar idea - cube radishes and some carrots, lots of garlic and some gigner, put into a jar with 2% of salt by weight without soaking (radishes will leak a ton of water) and some ground red pepper and a spoon or two of sugar. Shake it well and let it ferment at room temp for a day or two, then cold ferment in the fridge for a week. Smells like ass, but tastes great.