Check out is a commonly used phrase that is used in a variety of contexts to mean different things. Depending on the context it is used in, there are a variety of synonyms you can use instead. For instance, if you are referring to paying for something, then you may use words such as pay, settle, clear, or hand over. Alternatively, if you mean to examine or inspect something, words such as scrutinize, review, evaluate, or assess might be used instead. You could also use words like explore, examine, investigate, or look into if you are referring to exploring or researching something in detail. Whatever the case, there are many different synonyms you can use to describe the act of checking out.