By the way, you're not making a mistake leveling up all the classes at once. I did it that way and IMO it's the best, because the materials all inter-relate and support each other. Mats that you need on one job are synthed or gathered on another within the same level range. I advanced the jobs one by one 5 levels at a time, using multiple turn-in repeatable leves as the main way to do it.

At the lower levels, to HQ you need to use HQ mats.

When gathering, use your skills wisely to maximize HQ. At L15 gatherer, you get a skill that increases HQ probability. Use that skill on the +gathering nodes to maximize its effect. Also when gathering, choose only the 3 nodes that fit your goal and rotate around those. For example, +gather and +HQ if you need the HQ mats. Rotating only 3 will maximize the amount of boost you get in a gathering session.