There are many synonyms for the phrase "take a shot," meaning to attempt something or to take a chance. One synonym is "give it a go," which implies a willingness to try something new or risk failure. Another is "take a chance," which implies a level of uncertainty or risk involved in the attempt. "Make an attempt" is another synonym that conveys the seriousness of an endeavor. "Take a stab" and "give it a try" are also useful synonyms that convey a similar meaning with slightly different connotations. Ultimately, all of these phrases imply a willingness to try something new or take risks in pursuit of a goal.