There are several synonyms for the phrase "go out for" which means to leave one's home to do something. Some of the similar expressions to this include "step out for," "head out for," "venture out for," "leave for," "set out for," and "embark on." These phrases can be used interchangeably to communicate the idea that someone is leaving their current location to go elsewhere. For example, a person may say, "I'm going out for a walk," or "I'm stepping out for some fresh air," both of which suggest the same action of leaving their home temporarily.