Spore syringes can be stored at room temperature, out of direct sunlight, but they last significantly longer when placed in the refrigerator. Ideally, you would keep your syringe(s) in the original packaging and place in the refrigerator until you begin your research.
It is important to store yourmushroom spore printsin a sealed container and in a dry, cool location that is out of direct sunlight. The spores that make up the print can be exposed to contamination and activation if exposed to air exchange and humidity. If you live in a high humidity environment, prints can be stored in an airtight plastic pouch and in the refrigerator. Do not freeze your prints as this could damage the spores. Spore prints can remain viable for years is properly stored.
If you are beginning your research right away (within a week or two), room temperature, out of direct sunlight, is just fine.