The whole game revolves around buying gifts for the characters, creating a stronger bond, and proceeding in their storyline. You'll need to have 3 stars and a heart symbol on the gift icon for them to be ready to open up to you as stated in the game.
Unlike ATU, there isn't any 'you have to be at the specific location at a very specific time to trigger x', although there are a very few handful of times where character chapters only occur in a specific room or a specific time but that isn't too hard to figure out as they move around a lot.