Most plants will yield some colour, the most common (and consequently least desirable) is a yellow green/greenish yellow and brown tones and they may not be very fast to light or washing. Some plants produce more clear and durable colours and have become known as dye plants. Just as medicinal plants often have the second latin name of officinale, the ending tinctoria denotes well known dye plants.
All parts of plants can be used to dye. Many roots give great colour, like Madder and Alkanet. With some plants it's the leaves that are used, like Woad and Weld. Or it can be the flowers, as is the case with Dyers Chamomile and Lupin. Barks of shrubs and trees are also a source of colour, like the exotic Brazilwood and Logwood, or simply try Cherrywood prunings. Some mushrooms give great colour and of course Lichens. And then there are the insects, like Cochineal and Lac. The possibilities for experiments are endless.
Most of the time, one uses one part plant material to one part of wool (or other textile). The plant material is chopped up into small pieces and simmered (80-90°C/ 180-200°F), most often for an hour or so, to extract the dye. Flowers should be shimmered at a lower temperature or they tend to give browner colours and barks and tough root need a longer time. The dye solution is strained and cooled before it can be used to dye wool.
I tend to make the skeins 20-50 grams / 0.7 to 1.7 oz and tie them in 3-4 places. The wool is immersed in the dye solution and water added if needed. The wool needs to be able to move around freely in the dye solution.
One thing that is a bit surprising to beginners is that the intensity of the colour isn't determined by the amount of water in the dye solution, but by the ratio of dye material to wool (or other textile material). So to get lighter colours, one uses less plant material.
Since wool is very sensitive to sudden changes in temperature, it needs to be heated gently. It should take at least one hour to reach simmer. Most of the time the wool is simmered for an hour to get good saturation and to make sure the colour is fast. If the wool reaches the desired strength of colour sooner, it is best to continue to simmer it in clear water for the rest of the time to insure the best fastness. Just make sure the temperature of the water is equal to that of the dye solution. The used dye solution can be used to dye more wool until all the dye is exhausted from it. That way one can get a few increasingly light tones of the same colour.
Most of the time the wool needs to be mordanted before it is dyed. Without mordants, the dye won't adhere to the wool and simply wash off. The exemptions, substantive dyes such roots, do not need mordants, and barks contain tannins that are mordants. Lichen are also substantive as are Onion skins and Indigo. Mordants are most often metalic salts, the most common being Aluminium sulfate (Alum for short). Alum and Iron iron (ferrous sulphate) are quite safe to use, while the other frequently used mordants like Tin (stannous chloride), Copper (copper sulphate) and especially Chrome (potassium dichromate) are toxic and need care in handling. Each colour affects the final colour differently. Most dyers prefer to work with Alum. It is not toxic and give clear colours and good fastness.
The best pots to use are stainless steel. Pots made of Aluminium, Copper or Iron will affect the colours (very much like the mordants) so they can be fun to use. I have gotten most of my dye pots from thrift stores for very little money, so this doesn't have to be an expensive hobby. Substantive dyes do not require a dedicated set of dye pots and equipment so they are perfect for beginners.