Beginner weavers could go broke buying all the mats so here are some suggestions for the first 25 levels of weaver.
Moko Grass - unless you have Botany high enough you should just buy this from the vendor (2 gil each)
Cotton Balls - you can buy this but it gets expensive. I suggest getting Botany up to 11 and then you can farm cotton near Bentbranch Meadows. I got 5 stacks in phase 3 and it got me to level 14 botanist. Edit: 3 stacks of cotton balls + 1/2 stack of HQ cotton balls got me from lvl 11 - 14 in release.
Diremite Web - can't buy this but if you find it cheap on the MB then always grab some. I mostly would farm off of lvl 25 mites (Banemites?) in the North Shroud. Edit: You can get this from leves in Quarrymill. It's repeatable I believe but the reward is eventually replaced with crystals. (10 per leve I think).
Another tip: 5-6 levels in Leatherworking will save you on leather and hard leather. Or find a Leatherworking buddy and trade Yarn for Leather.
If you know other good sites or tricks LMK!
Edited, Aug 27th 2013 1:52pm by algorithm0r
Edited, Sep 4th 2013 1:31am by algorithm0r