The most important foundational documents for public libraries is the bylaws.
If you are a statutory city (most are), Bylaws are the rules for how your library board conducts business. In the bylaws it should be specified how the board is comprised, how board members are selected, what are the terms of board members, and when elections are to be held. The bylaws should, ideally, specify the library director is the non-voting executive officer of the board. Many variations are possible, and much more might be included. For more information, read your bylaws to see if they make sense, then seek assistance from your regional library system, and/or your attorney. Bylaws should be reviewed periodically, at least every 5 years, to be sure they remain current, and can generally be changed and updated by the board.
If you are a charter city, the city charter spells out how the city intends to do business. If the library governance is addressed in the charter, that serves to guide how your library is governed. If the library is not addressed specifically in the charter, your library defaults to a governing board under statute 134, and the rules are the same as they would be if you were not in a charter city. You need to understand your charter, what it says, and consider consulting an attorney.
If you have questions about your city, you may contact the League of MN Cities (or Steve or Krista).
In regard to other foundational documents, State Library Services asks on their annual report form if your library has the following in place, a strong indicator they consider these to be important:
Records Retention Schedule
Building Accessibility Plan
Internet Acceptable Use Policy