Most of the side quests are unlocked by progressing further through the MSQ, some only becoming available upon completion. Some require the completion of all side quests in the area. For instance, the cave questline in Hinterlands only continues after completing all broom quests. There are a few quests also tied to the completion of the moogle quests in Churning Mists. So my advice is to look for any available quests in Ishgard, as these are often the start of cutscene quests. Hop on a job you don't have 60 and go wild. There's a lot of quests...
