It is not easy to decide for a non-native speaker, so I want to ask which one is the right one:
"To ask someone for a date" or "to ask someone on a date"?
Or are both of them correct?
The two definitely have overlapping meanings, and I think which is more common may depend on dialect. To me, as an educated British person who's been exposed to a lot of American culture (haven't we all?), the difference in meaning is very subtle, and reflects some psychology.
"To ask someone on a date", I would expect to see where the person doing the asking has some relatively specific idea. It suggests confidence, though not necessarily excessive confidence. "To ask someone for a date" suggests the attitude of a supplicant, as if the other person has something you want, and you hope they deign to give it to you.
(Note that either may have different meanings outside the context of 'dating', i.e. romance; "to ask someone for a date" particularly can have uses other than this.)
In my experience: To ask someone on a date means to invite someone to go with you on a romantic excursion.
To ask someone for a date could mean the same thing. However, I hear that construction much more frequently when discussing scheduling. "We said we'd meet in February to discuss our project. I asked you for a date, but you still haven't gotten back to me."