Stainless steel is very low maintenance and can be easily cleaned with soap and water. No special storage is needed, just avoid wearing your jewelry in chlorine as chlorine can break down the outer layer of stainless steel that protects it from corrosion.
The moonstone in your earring is made for everyday wear and built to last, but I do recommend being mindful while you wear your piece. Please be cautious not to drop your piece or knock it against hard surfaces to avoid cracking or chipping.