Where do I go to move on with the story after getting the gem from the volcano? I have no prompt, and have finished all missions that I can.
Well, you go into the volcano by going north-east, where you fight the next First One. If you mean after that, then just go far east from where the volcano is, then north. The area should start to look snowy, then go into the first town you see and the story should tell you where to go next. Hope this helped. 
It said something about the Kings kidnapped daughter. But I have no story prompt.
Oh, well you just have to go west and when you see that cave after you go far north, enter it. The princess is in there held hostage
A henchman just comes up and says get out and my character walks away
That is strange… It has been a while since I played my story mode, so can you tell me the last few story events you did before this.
And I think you were supposed to initiate a fight with the henchmen… Then you go into the cave