I have trouble knowing when to use "için" and when to use, for example, "bana" in a situation like this. I know some of this depends on the verb, but in a case like this, could you have said "benim için" instead of "bana"?
Yes, it would have the same meaning.
Please, where to put (on sentence in a situation like this) "için" and "bana" .
Do I assume that Anneler gunu is a standard expression? I ask because this is a plural but in the English it is singular.
Yes, in Turkish the standard expression is Anneler günü.
Actually in my family we are careful to put the apostrophe after the s, because the day belongs to all mothers: Mothers' Day.
Is annenin günü wrong?