There are lots of programs at the Kennedy School - 4 master’s and 4 doctoral - so the answer is going to vary depending on which one you want.
Other than that, HKS has lots of general information on their website about these issues.
Courses: Here (https://www.hks.harvard.edu/admissions-aid/masters-program-admissions/how-apply/degree-program-prerequisites). Each program has its own required/recommended courses.
Letters of recommendation:
*If you are an undergraduate or have less than two years of work experience, we encourage you to submit two letters of recommendation from university faculty members—one from your departmental major—and the third from your immediate supervisor from a professional work environment or internship.
If you have more than two years of work experience, two of your three letters of recommendation should be from current or previous work colleagues, including a supervisor. We encourage you to also submit an academic reference.*
CheddarcheeseMN is right in that all Princeton Woodrow Wilson MPP students get tuition remission and a stipend; however, that program is a mid-career MPP, and they require at least 7 years of work experience in the public sector.