Generally, Probate and Family Court handle custody and visitation and most parenting decisions. This can be involved on divorce cases, separate support cases, paternity cases; where the parties are not married, adoption at many times and other issues.
Both Probate Court and District Court along with sometimes Juvenile Court can handle a restraining order or 209A orders which can also have custody or parenting plans involved. If it involves DCF, then Probate Court occasionally can handle that, but Juvenile Court would handle those cases more often.
Juvenile Court and Probate Court can handle Guardianship cases. Grandparent Visitation cases are generally heard in Probate and Family Court, but if there is an active case in Juvenile, it can be heard in Juvenile Court. Also, what used to be called CHINS cases, now are called CRA cases, and Delinquency cases are handled in Juvenile Court.