The library is unlocked after completing a long quest while working in the guild. That minigame where you send people to places based on their strengths and weaknesses. You need to complete the one about reclaiming a great library, I believe.
Then Luna will give you a new quest. For spoilers sakes, I cannot say more.
The other one, I'm unsure of. But it says it has a "hidden condition" and I feel like it might be related to another quest which randomly appeared for me after completing the guild minigame a bit.
So just, work at the guild a bunch? I guess. Seemed to make new shit show up for me.