Email Introduction by a Mutual Contact: Your mutual contact can give a nice introduction to you and propose a meeting to talk about your goals or interests.
Follow-up: Send a separate email, or reply to the original, thanking your contact for the introduction.
Expand on the Introduction: Give your new contact a little more information about yourself—your background, your goals, and why you wanted to meet them.
Request for Informational Meeting: Write a few sentences about your intentions for meeting and provide a few dates and times you could meet, but also let them know you are open to other times if the ones provided are inconvenient.
Informational Meeting: Make sure to have a few key questions ready to ask the person. What path did you take to arrive at your contact position? Did you receive assistance in building your network? What would you suggest I do now to get to X? Do you know of any offices or programs looking for assistance where I can get the firsthand experience and build my resume for X ultimate position?
Introduce Yourself: Mention a common factor such as school, leadership roles, geography, or hobbies. You want to build a rapport. Let them know how you learned about them.
Share the Reason for Contact: Tell them you are interested in learning more about their path to a certain position at work, a nonprofit, or an appointment.
Of course, it's an advantage if you already have a connection in common. But that doesn't mean you are out of the game if you have to spark the conversation yourself. Your proactive attitude of setting up an informational interview will be your best chance at getting the person to agree to meet you. Good luck!