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First things first: Don't you dare wash them until you're ready to eat them or use them strawberries are like small sponges, ready to soak up all the water they can come into contact with, and once they've soaked it up they are quicker to turn to mush and rot away.
Second: If you plan on eating or cooking with the berries within a day and it's not too terribly hot in your kitchen, you can leave the strawberries out at room temperature. For overnight storage, however, you're better off refrigerating them.
Best case scenario: Line a shallow bowl or rimmed plate with several layers of paper towels or a clean kitchen towel, place the strawberries in more or less a single layer on the towels, cover, and chill the berries until you're ready to use them. Stored this way, very fresh strawberries will keep for several days. The closer you can create this dry (the paper towels soaks up excess moisture) and un-pressed (single layer) situation, the better.
For more than a few days: If you're not planning on using the strawberries within a few days, you're better off freezing them than trying to keep them all fresh and unblemished. Frozen strawberries are perfect for whirling in smoothies, turning into sauces, or baking up in pies, tarts, cakes, and other treats.