bother has 28 distinct synonyms spread across 5 of its 8 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.
take the trouble to do something; concern oneself
“He did not trouble to call his mother on her birthday”
to cause inconvenience or discomfort to
“Sorry to trouble you, but...”
intrude or enter uninvited
“Don't bother the professor while she is grading term papers”
make nervous or agitated
“The mere thought of her bothered him and made his heart beat faster”
make confused or perplexed or puzzled
something or someone that causes trouble; a source of unhappiness
“washing dishes was a nuisance before we got a dish washer”
Synonyms, antonyms and the broader and narrower relations for bother come from Princeton University’s WordNet. Full sources and licences.