The Goddess Festival is a pretty huge festival. It usually gives a pretty big affection boost to the girl you bring. To ask a girl you have to ask her on spring 7th. I found that if you are under a blue heart the girl will say no, however, no matter what i did in year one they won't say yes. Therefore, this is a year 2 on festival. If you ask the girl they will show up say yes and the following will happen:
Afterwords the girl will ask you to go somewhere, say yes to increase the affection towards you. However if you just leave her at your house and forget her the following will happen and you will lose affection: