phrases - Me and X or X and me?

When should I say, for instance, "Mary and me," and when should I say "Me and Mary?"

Example:

Which option should I use in the following sentence?

After drinking our tea and saying goodbye to Hank, [...] made out way back
to the hotel.

Answer

The order is not a matter of grammar but of convention. It is generally thought to be more polite to mention the other person first.