Tools, Supplies, and Safety
I have called this section "Tools, supplies, and safety", but I really should have included "environment" as well. Some of the chemicals we use can be dangerous to people, but also to the planet. Some of the metals we use to mordant will concentrate up the food chain, or contaminate water (particularly if you're on a well!). Some of the dye stuffs you will read about can be endangered species. This is one of the reasons I don't dye with lichens; they are a diverse species and some of them take hundreds of years to recover. Although I know it can be done responsibly, I have chosen to focus on readily available dye stuffs that I buy commercially and grow myself... and occasionally that the neighbours unknowingly grow for me.
Water. Dyeing is a very thirsty hobby. You will need plenty of water to do natural dyeing.
Pots! Whatever you are dyeing will need to be able to move freely in the pot, so the larger the better. Beware that larger pots get very heavy when you fill them... I may have had to scoop water out so I could get it to the stove.
Rubber gloves... trust me on this. Yellow dyes make you look jaundiced, red dyes make you look like put your hand in a blender, and blue dyes make you look like a zombie. In all seriousness though, some dyes and mordants can cause reactions and it's best to always wear rubber or other waterproof gloves.
Spoons... you know, for stirring.
Sieve, for removing plant material that will otherwise leave you with sediment and/or splinters in your fibre. Along these lines, a loose weave linen or cotton cheesecloth is useful for the straining. With many natural dyes both linen and cotton don't dye well, so it doesn't remove much of the dye from the vat.
Something to heat your dye pot on. I will use my kitchen stove in the winter, but in the summer and spring I have a heating element I use in an easier location (ie. one that I don't need to worry about staining. I know they told you your sink was stainless steel...It will also splatter on your walls, your stove, your countertops, your floor, your furniture... you get the idea)
Candy thermometer (some dyes aren't really heat sensitive but wools and silks can be damaged if the dye is too hot). If you're dyeing without a thermometer, which I often do because I lose it under piles of yarn, a good rule of thumb is, if you can put your hand in it without burning yourself, you're probably good... that being said, don't burn yourself and remember your gloves.
Something to dye! Natural dyes only don't work well (or at all) on synthetics. They work best on silk and wool (the protein fibres. Linen and cotton can be dyed with some dyes, but not all.
Plant material vs. extracts
Extracts are easy. You put them in the pot, they dissolve, you're ready to go!
Plant materials require more steps. They often need to be soaked first, then heated in water on the stove, then strained, then possible sieved again. These steps aren't onerous and you often get a better colour than with the extracts. Particularly, reds vary a lot, so extracts don't always give you the full range.
Mordants and other chemicals
Most natural dyes require a "mordant" to allow the dye to "stick" to the fibre. The mordants also affect the colour of the dyes. The most common, and safest mordants are alum and iron sulphate. Alum will make the colours brighter. Iron will make them duller, or often completely change them. For example, marigold is a bright yellow with alum, but a khaki green with iron.
Other chemicals that can shift the dye colours (particularly in the reds) are cream of tartar, soda ash, and vinegar.
In order to dye using indigo or woad, you will need reducing agents such as lye and thiourea dioxide.
With regards to the safety and use of various mordants, I recommend this thorough description here.