You've Been Storing Fresh Herbs Wrong Your Whole Life
Often when people buy herbs at the grocery store, it's simply for the use of one or two sprigs and then the remainder languishes in the crisper drawer of the refrigerator for a few days until they're deemed to be past their prime and then get thrown out.
However, herbs can stay fresh in the refrigerator for weeks if they're properly stored. Of course, this involves a little work taking the herbs out of their store packaging and preparing a different storage method, but it'll be worth it when you're not buying basil every week at the grocery store.
Leafy and delicate herbs like basil, cilantro, and parsley should be treated as if they're flowers. Snip off the bottom of the stems, and put them in a jar of water with a plastic bag wrapped (or rubber-banded) snugly around the whole thing to keep evaporated water inside and making the whole environment moist so that the herbs don't dry out (via Simply Recipes). While parsley and cilantro can be stored in the refrigerator, basil is best kept at room temperature. When kept this way, these types of herbs can last for at least two weeks.
How to best preserve low-moisture herbs
For other types of herbs with lower moisture content, such as thyme and rosemary, you'd be better off wrapping them in a damp paper towel and then sealing them in a plastic bag (via Spiceography). However, thyme can also be stored in the same glass jar method way as the herbs mentioned above, and can also keep for weeks this way (via Spiceography).
A third option to keep your low-moisture herbs (like rosemary and thyme) usable, if not fresh, is to dry them if you don't think you'll use them in less than a few weeks. Hanging them upside-down in a room with good air circulation for a few weeks is an easy reliable way to dry them, extending their usefulness by a few months at least (via The Spruce).