There are many good books with formulae galore, so I will not go into great detail. Here is, however, the recipe I use for experiments:
Fiber - #amount
Dye material - #amount
Water - to cover
Simmer one hour, cool overnight in dyebath. Rinse well in cool water.
These samples are the result of dye experiments by myself and my fiber demonstration partner and longtime fiber artist, Dhar Benker. We use these samples to demonstrate the range of colors available dyes from "natural" (non-synthetic) sources. The demonstrations we paricipate in range from Medieval Europe to the American Civil War, so we like to choose dye plants that would have been available in Western Europe and America from 600 to 1860 C.E., prior to the advent of synthetic dyes.
The fibers used in these experiments and demonstrations are primarily on wool, but we sometimes use silk or cotton.
O - no mordant.
Al - pre-mordanted with alum.
Fe - mordanted in the dyebath with iron, either dyed in an iron
pot or in a non-reactive pot in water steeped with rusty nails.
Sn - tin, mordanted in dyebath using a handful of pewter coins,
or water that had pewter coins steeped in it or several days.
BS - baking soda after rinse
Am - ammonia after rinse
V - vinegar after rinse
C - cotton fiber
S - silk fiber
Above, L to R: undyed wool,
white (O); "staghorn" lichen, unidentified species,
pale clear yellow (O); calendula, fresh flowers, pale dirty yellow
(Al); vetch (Lathyrus sp.), fresh flowers and leaves, pale
cream (Al); bracken fern, unidentified species, pale pinky beige
(Al); calendula, dried flowers, blah beige (O); Bermuda sorrel
(Oxalis pes-caprae), bright yellow darkening over time
to bright gold (O); nasturtium, fresh flowers, blah beige (Al).
Above, L to R: yellow onion skins, common grocery store variety,
golden tan (Sn); Usnea barbata lichen, tan (O); curly dock,
mature flowers (Rumex crispus), tan (O, S); yellow onion
skins, tan (O); yellow onion skins, golden tan (Al); yellow onion
skins, golden tan (Fe); bracken fern, unidentified species, deep
golden brown (Al, Fe, BS); bracken fern, unidentified species,
golden brown (Al, BS); sassafras root, dried, "redwood"
brown (O); Usnea barbata and bedstraw root (Galium aparine)
overdyed with sassafrass, "redwood" brown (O).
Above, L to R: Usnea barbata overdyed with walnut, dried
& powdered hulls, pinkish brown (O); black tea leaves, deep
redwood brown (O); black tea leaves, deep redwood brown, slightly
darker than without ammonia (O, Am); unidentified "staghorn"
lichen overdyed with Bermuda sorrel (Oxalis pes-caprae),
then overdyed with sassafrass, chocolate brown (O); coffee, greenish
tan (Fe); yarrow, dried flowers, greenish blah beige (Al); Scotch
broom, fresh leaves and flowers, greenish blah beige (Al); rosemary
(Rosmarinus officinalis), olive green (Fe); rosemary (Rosmarinus
officinalis), clear olive green (Fe, V); rosemary (Rosmarinus
officinalis), dark olive green (Fe, Am).
Above, L to R: alder bark, fresh, olivey brown (Al, Fe); indigo
vat (O); privet, garden variety, berries, fresh, dusty blue (Al,
Fe); red cabbage, common grocery store variety, bluish gray (Al);
bestraw (Galium aparine) fresh root, pale piny beige
(O); elderberry, leftover from making elderberry liquer with sugar
and alcohol, pale pinky beige (O); walnut, dried & powdered
hulls, pale pinkish coral (O); blackberries, fresh fruit, pinkish
tan (Al); red plum leave, pinkish beige (Al, Sn); natual gray
wool, undyed; natural black wool, undyed (it's there in the shadow,
really!).