To become close friends with Hagrid you have to complete side quests. The ones I have finished you can see here.
Side Quest: Fairytale
Hagrid is keeping fairies in his garden. Snape wants to have their wings for potions. So Hagrid wants to get the fairies to a save place. You are going to help him. In the magical creatures reserve you can adopt a fairy. She will also help you and Hagrid to rescue all the fairies from Snape. You can see the whole story in this YouTube Video.