Once you can enter the Labyrinth, do so. A 2 day (iirc) timer will start as soon as you do so. Till the end of this cooldown, you can explore the labyrinth. At the end, you will get all items in your satchel.
Your Dragon needs to be in your city, or else you can't enter the Labyrinth.
Use Dragon Spiritresearch at the University to increase your Dragon Spirit stamina, health, potions, etc.
Once you reach floor 11, 21, 31, etc, you can start at that floor next cooldown.
Try to reach as many floors as possible by going down as soon as you kill the guardian. Your satchel usually is full before your stamina runs out anyway, at least after some research. The further you progress, the better loot.
Satchel capacity usually is more important than slots.
Check your Dragon Spirit Equipment Scrolls at your Spirit Chamber to decide which materials you need most.
You can pick up wood/food as you explore the dungeon, but I'd advise you to discard them as soon as you need satchel space. It's far easier to gather those than the items needed for Dragon Spirit equipment.
Leveling up your Dragon Spirit is quite important, as it raises its stats, so Dragon Spirit EXP might be the most important item to pick up.