Close synonyms include head home. Each sense on this page lists its own synonyms, because a word with several meanings rarely has one substitute that works for all of them.
Is go home a common word?
go home is banded "Familiar" — widely understood, used when the subject calls for it. The band comes from how often the word appears in a large corpus of everyday English, not from an editor's impression.
Definitions, synonyms and semantic relations for go home come from Princeton University’s WordNet. Commonness is measured against a corpus of everyday English. .