Spider silk has many good places (East Commons, Butcher Block, Misty Thicket, Oasis of Marr, N Desert of Ro, etc..). Both large and giant spiders drop the silk, hence there appears to be more mobs that drop them.
Spiderling silk is a little tougher to get. Although there are plenty of spiderlings in EC I've found these to be heavily camped. The newbie area in the N Desert of Ro seems to be pretty good (both in the green area by the Inn and the early sand portions). Both desert spiderlings and dune spiderlings drop the silk. If you're very low level be careful of the dune spiderlings, for they have a poison bite.