Right, but as of right now, she hasn't given me any quests, and I can't even try increasing the relationship through gifts or anything. What quest do I need to do to unlock those options so I can start doing all that?
What options do you have when talking to her? The quests are under the gift. You may need to do the Rank Promotion quest first. Try that.
To do the relationship events you must go talk to her while you are alone