How do I bring the Unusual Gem to an appraiser?
One of the things you can do is show it to Maul, the Nord who accosts you when you first enter Riften. He will direct you to Vex, in the Thieves Guild, who starts the next quest, No Stone Unturned.
You cannot access Vex without joining the Thieves Guild, so if you're interested in completing the quest, you'll have to join up.