There isn't one way to store all of your fresh herbs because herbs themselves fall into two categories: hard and soft. Food52 explains the difference and suggests the right method to keep each type of herb fresh for a longer time.
Hard herbs are the ones with woody stems, like rosemary, oregano, marjoram and thyme. Soft herbs are those with tender stems, like basil, parsley and coriander.
Soft herbs like to be treated as you would a bouquet of flowers. Snip the base of the stems and put them in a glass of fresh water, changing out the water every day or two if it starts to cloud. Store them in the fridge.
Store your hard herbs by wrapping them first in a damp paper towel, then loosely in plastic wrap, and keeping them in your crisper. (If you're going green, a damp kitchen towel and airtight container works perfectly well.)
Remember not to wash your herbs until they are going to be used. And basil is the only herb you shouldn't be keeping in the fridge — it's best left at room temperature.
The Best Ways To Store Fresh Herbs [Food52]