Check your achievement log. Daily quests and any achievements will show in that list. Daily quests reward 20 loyalty points, so they will add up. You also get loyalty points from greeting friends (10/day), loyalty token drops from bosses, loyalty points from killing certain bosses, etc.
Also, the fragments are meant to be a way to supplement completing a key. I wouldn't buy those fragments until you are close to finishing a key. It would suck to buy a bunch of fragments and then get a whole key to drop.