What's the best product to use to polish up a sterling silver moon stone ring?

I use a silver polishing cloth. There's no need for water or acid based liquid cleaner. Moonstones can't be immersed in liquid cleaners and abrasive cleaners scratch the delicate surface of the moonstone. A polishing cloth shines up the silver and leaves the stone alone. I got my cloth from a jewelry supply store but you can easily find them on Amazon. Enter silver polishing cloth into Google and you'll find even more place to buy one

Yes silver polish, but avoid the stone. Put it on the metal and rub it a bit to get minor scratches out, then let it dry on there for a while, before buffing it up.