As stated, dungeons will be required to progress the story, as will a couple of instanced battlefields, as I recall. You'll also have instanced battlefields with your job quests. DPS queues are long because there are more DPS than there are tanks or healers. Though, there are a few dungeons out there where DPS gets to do a lot more than the tanks or healers, like picking up adds, manning cannons, and some other mechanics.

I would recommend doing fates and levemetes while waiting on dungeons, as these help you level and in the case of leves, you can get different rewards like ventures, crafting materials, and gear for equipping, selling, or desynthing.