Please explain which of the following (or any other) is more appropriate to say "enter postgraduate school/studies":
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「入る」and「入学する」mean entering school. Therefore, you can use them regardless of what kind of school. In addition,「入る」is more informal than 「入学する」and you should use the latter in writing.
On the other hand, if you use「進学する」, you will lay stress on entering "more advanced" school. For example, if you go on to graduate school directly after graduating college, No.3 is the most proper expression.