Synthweaving is powered by Archaeology.
Archaeology is a gathering profession. On your minimap, you'll see nodes, which you can then pickup (or if you're feeling dictatorial, you can send your buddy to gather them instead while you go pull the next group). Trivia: if you send your buddy to gather a node, they'll freeze in place if you then start an NPC conversation.
Archaeology levels up via gathering, so spending credits on missions is not necessary. I went from 1-100 archaeology just by keeping an eye on my minimap while levelling. Note that, just like herbalism and picking peaceblossoms, you stop levelling up. You can still send your companion out on missions to get stuff, in which case you're spending credits to buy skillups and mats.
From this, I then could power up Synthweaving. Synthweaving makes Light, Medium, and Heavy armor that is (probably?) Jedi-specific -- i.e., not suitable for Troopers and Smugglers. (I'm not sure what stats they want, and I don't think the armor 'weavers create has Cunning on it, for example.)
* When you mouseover a node, it'll tell you what it is.
* Run over small children to get to Lost Fragment nodes.
* Run over small animals to get to colored nodes (amorphous red crystals, green crystals, etc).
* Getting to rubat crystal formations is nice.
Synthweaving has levels of mats, so similar to linen/wool/silk/runecloth, etc. In the beginning, you only need rubat crystals, which are the easiest to find. Later, you're limited by lost fragments.
By reverse engineering what you create, you can discover upgraded schematics. Most if not all require stuff that I think you can only get from Underworld Trading. (I've never found Shadow Silk, that's for sure.)
Once you're into Level II synthweaving, you'll need some very cheap vendor mats. There's a crafting vendor in the east wing of Vendor Alley on Coruscant who sells something for 10 creds a pop that I often need 2 of per item, for instance.