If you have trouble recognizing which of the forms to use, determining whether it's object or subject, simply remove "My father".
I went to the cinema.
My father and I went to the cinema.
or
The guards stopped me at the border.
The guards stopped my father and me at the border.
It works the same, no matter if you're at the end of the list, in the middle of it, or without a list, just alone - and you should know how to handle that last case.
You will sometimes see people writing "My father and me went to the cinema.". That's common, but that's wrong - it's a very common error. Don't repeat it.