There are a variety of synonyms for the phrase "go back and forth." For example, some may use the term oscillate to describe a repetitive movement between two points. Alternately, others may use the term shuttle to describe a similar movement, often in reference to transportation or communication. Some may use the phrase move to and fro to describe a similar motion, while others may prefer to use the term sway. Additionally, one may use the phrase pivot back and forth, or even waver to describe a hesitant movement between two choices or ideas. Ultimately, the choice of synonym will depend on the context in which the phrase is used and the desired tone of the communication.