For plant-based cloth products, you need a pig tail farm, a farmer's workshop, and loom, and either a craftsdwarf's shop or a clothier's shop. The farm produces pig tails, the farmer's shop turns the pig tail plants into pig tail thread, the loom turns the thread into cloth, and the other 2 shops turn the cloth into finished products. The craftsdwarf's shop produces cloth crafts, while the clothier's shop produces actual clothes, ropes, and cloth bags.
To turn pig tails into pig tail thread, you run the basic "process plants" job at the farmer's workshop.
You cannot make the process fully automated without extensive modding, including several custom workshops and custom reactions. You can, however, produce more pig tails than your farmer's shop(s) can process, process more thread than your looms can weave, and weave more woven cloth than your clothiers can cut into finished goods. This will allow you to simply set all the jobs on repeat and forget about them. Alternatively, you could play with DFHack's workflows tool; I've not tried it myself, but I understand that this sort of thing is what it was made to do.
The only thread-plant that can be grown in spring and winter is the rope reed above ground.
Optional parts of the cloth industry involve setting up a dimple cup (or other dye plant) farm, a millstone or quern, and a dyer's shop to produce dye and turn dye and thread or cloth into dyed thread or cloth. This adds value. It's often convenient to have your pig tail farm produce dimple cups during the off-season so that plot is always producing. You can set standing (o)rders so that clothiers use only dyed cloth.