Knowing how to make your own pigments is a great thing for painting and dying things.
The process of making pigments is pretty simple. Find something that dries the color you want, dry it very slowly(The slower you dry it the more color will be preserved) then grind it into a very fine powder. For grinding you're going to want something smooth, flat and nonporous. A thick bit of glass works, but a stone that fits probably wouldn't be hard to find. Now for the grinding stone, you still want smooth and nonporous, but go for something with a bit of a curve on it so you can rock it around. Start with small bits of what you're going to grind and work it into a very fine powder. If you Start with something too big it will be hard to break up and grind into small particles. Once you have a fine powder gently dump it into the container you wish to hold it in.
Now, knowing how to grind something up isn't very useful if you don't know what makes colors. Here's a list of a few different things to start off with, but experiment, try to find different colored things and grind them up.
Black - Black pigment is almost always made from charcoal ground up. Sometimes cooking bone over a fire is used too, this gives a slightly different tint of black. Burning coal or tar work as well too.
White - Chalk or Gypsum work great for whites. Bone and horns give a slightly off white to yellow color as well.
Red - Traditionally red is made from the dried bodies of Dactylopius coccus but there are other options if you don't want to chase bugs around. Resin from some palm trees work great, but for ease of use go for any red fruit or berry. They almost always make nice reds.
Blue - Blue pigments have always been rare and hard to come by. Ancient Egyptians were famous for "Egyptian Blue" one of the first blue pigments that involved mixing copper, lime, sand and natron, then baking it in a furnace. Now since this will probably be hard for you do to, your best bet is indigo leaves or duck poop. But really, there was a reason most ancient art didn't have blue.
Yellow - Yellow is made from dried urine, yellow clay, or a few different berries. Take your pick on what you want to gather.
Green - Most plants and unripe berries can be dried and turned into different shades of green.