walk around has 3 distinct synonyms spread across 2 of its 4 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.
walk with no particular goal
“we were walking around in the garden”
behave in a certain manner or have certain properties
“He walks around with his nose in the air”
walk randomly
“We were walking around in the neighborhood to see whether we could find an open drugstore”
Synonyms, antonyms and the broader and narrower relations for walk around come from Princeton University’s WordNet. Full sources and licences.