Involvement in student organizations—particularly leadership positions—is a good way to gain experience in project management and working with teams, and presents an opportunity to learn alongside other graduate students how your skills as a PhD student transfer into other domains. Student groups generally involve a reasonable time commitment, and cover a wide array of interests. Consider joining one or more groups as a member and then, if and when the opportunity arises, move up into a leadership position.
One of the best opportunities for involvement in student groups is to join GSA or GPSS, the two major graduate student government bodies. Members of these groups get the chance to work with students from around the university and also to interface with administrative offices and the City of New Haven on issues of interest to graduate students, including transportation, childcare, and more.