When you start your journey in the New World Weaving with lvl 0, you are able to harvest plants for Fibers and refine them into cloth, in particular, Linen. So take your Sickle and go for tons of Hemp. You can refine them at the station in the settlement called Loom. 

Looms come in different levels in accordance to how Guild developed their settlement, much like the crafting stations. So to craft a particular cloth you may look for a particular tier Loom in another settlement.

As for leveling your New World Weaving skill, it will rise accordingly to the amount you use it. The more you refine, the higher your level will be, so you need a lot of resources to upgrade. 

Then you get access to more complicated cloth like Sateen, Silk, Infused Silk, and Phoenixweave. They require a previous level cloth plus refining materials to refine. You can find them either in a supply container, trading post, or craft by the Material Converter. The Material converter should be bought every time and can only be used once.

You also have an option to grade your cloth down if you need a particular ingredient. For this, you need to obtain Weak Solvent. 

When you get to the point of crafting high-level Silk and Sateen, you may consider a Material Converter for Cloth Weave. They come in different levels too.