There are multiple tiers of processing. Your basic skill allows you to heat, chop, thin, dry, shake, and grind raw mats into refined materials. If you want to take those refined materials and refine them further, you either need a corresponding workshop, or the knowledge to do it yourself. There are three tiers of knowledge for each of the 6 processing types (beginner/skilled/professional). Here is how you discover them.
The chain of quests stretches back a ways but essentially when you come to Serendia you will be directed to Alejandro Farm. Do all of the quests there and you should eventually get to In Search of Higher Processing Knowledge... which will take you to Ficy in Heidel.
At this point you will need to have Gathering Apprentice 3 to proceed.
You will get Learning Higher Processing Skills which will send you yo three other NPCs to learn your first Beginner tier skills.
Grinding: Beginner - Ornella
Materials for Equipment Repair
Chopping: Beginner - Lebyos
Working the Ash Tree
Heating: Beginner - Flaviano
Pure Water for an Experiment
The other skills are also available around Heidel at this time:
Drying: Beginner - Techthon
Need Soft Hides!
Shaking: Beginner - Lara
Toys (?) for Kids
Thinning: Beginner - Dora Fonti
Now you can process your melted shards into ingots, your planks into plywood, your soft hides into fine soft hides, and so on.