Cover the infusions with an airtight lid and place them in a room temperature, dark storage space. As long as the alcohol is covering the infusions, it will prevent mold from forming, even when stored at room temperature.
Each infusion needs to steep for various amounts of time. Some stronger flavors like citrus or peppers infuse quickly (24 hours or under), while others like dried herbs need a longer steeping time (7 - 14 days).
I'd suggest checking your vodka infusion every 12 hours (in the morning and at night) to taste the flavor. Since every ingredient will need a different amount of steeping time, it's best to check each infusion separately until you achieve a flavor you're happy with.