draw has 51 distinct synonyms spread across 27 of its 45 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
make, formulate, or derive in the mind
“I draw a line here”
select or take in from a given group or region
“The participants in the experiment were drawn from a representative population”
elicit responses, such as objections, criticism, applause, etc.
“The President's comments drew sharp criticism from the Republicans”
move or go steadily or gradually
“The ship drew near the shore”
bring or lead someone to a certain action or condition
“She was drawn to despair”
write a legal document or paper
“The deed was drawn in the lawyer's office”
engage in drawing
“He spent the day drawing in the garden”
move or pull so as to cover or uncover something
“draw the shades”
allow a draft
“This chimney draws very well”
require a specified depth for floating
“This boat draws 70 inches”
pull (a person) apart with four horses tied to his extremities, so as to execute him
“in the old days, people were drawn and quartered for certain crimes”
contract
“The material drew after it was washed in hot water”
reduce the diameter of (a wire or metal rod) by pulling it through a die
“draw wire”
steep; pass through a strainer
“draw pulp from the fruit”
flatten, stretch, or mold metal or glass, by rolling or by pulling it through a die or by stretching
“draw steel”
a gully that is shallower than a ravine
an entertainer who attracts large audiences
“he was the biggest drawing card they had”
a playing card or cards dealt or taken from the pack
“he got a pair of kings in the draw”
a golf shot that curves to the left for a right-handed golfer
“he took lessons to cure his hooking”
poker in which a player can discard cards and receive substitutes from the dealer
“he played only draw and stud”
Synonyms, antonyms and the broader and narrower relations for draw come from Princeton University’s WordNet. Full sources and licences.