Rinse fresh herbs well and lay on a paper towel. Wrap loosely in a paper towel and place in zip lock bag. Store your bag of herbs in your refrigerator's crisper.
If you won't be using your herbs within a week or so, its best to freeze rather than drying. Frozen herbs, once thawed, can be used just like fresh herbs in any recipe, though they will lose some of their flavor over time, so its best to use them within a year and use a bit more than a recipe calls for. Here's how to freeze fresh herbs:
Remove leaves from stems and place in a zip lock bag, spreading the herbs out, so there is a single layer. Lay flat in freezer.