Don't forget to take the greens off your carrots and beets to avoid limp roots.
Basil is sensitive to cold temps and will turn black if refrigerated. Strip the lower leaves of basil stems and put the stems in a glass of water on the counter to keep basil fresh. Wrap the stripped leaves (or all your basil if your fresh basil arrives w/o long stems) in a dry paper towel and keep in an airtight container at about 50 degrees or room temperature.
Most produce stored in the fridge will benefit from a plastic bag, greens bag, or other container to slow respiration rate and avoid limp leaves.
Green garlic and garlic scapes should be stored in the fridge but garlic bulbs belong in the cool, dark pantry.