"Go one's happy way" is a phrase that refers to moving on or resolving a situation in a contented manner. There are several synonyms that can be used in place of this phrase, including "move forward joyfully," "progress cheerfully," or "proceed happily." Other similar phrases include "walk on sunshine," "march confidently," or "saunter blissfully." All these synonyms evoke a sense of positivity and optimism, indicating the speaker's success or fulfillment in a particular situation. Ultimately, the choice of synonym will depend on the specific context of the statement, as well as the speaker's intended tone and message.