do has 28 distinct synonyms spread across 15 of its 16 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.
behave in a certain manner; show a certain behavior; conduct or comport oneself
“You should act like an adult”
travel or traverse (a distance)
“This car does 150 miles per hour”
Synonyms, antonyms and the broader and narrower relations for do come from Princeton University’s WordNet. Full sources and licences.