Question: As soon as I get home from the grocery store, I always put my tomatoes in the crisper section of the refrigerator. Is that the best place for them?
Answer: No — in fact, you should always store whole tomatoes at room temperature until they’re fully ripe.
As the U.S. Centers for Disease Control and Prevention notes, storage conditions below 55° Fahrenheit (like you'll find in your fridge) damage the quality of tomatoes. At those chilly temperatures, the flavor-producing enzymes that are normally present as a tomato ripens stop working. And you wind up with a less tasty tomato.
Once your tomatoes are fully ripe, you can extend their shelf life for two to three more days by placing them in the fridge. Try to bring them back to room temperature before serving.
After you've cut into a whole tomato, you should refrigerate any leftovers right away. As noted here, they'll keep well for a few days in the fridge; you can also freeze them for later use in cooked dishes.