Gender roles would be one theme. That theme is illustrated through Pernese society. The main character is a female, and she is discovering throughout the novel that Pernese society undervalues women. They have fewer opportunities than men and are given less responsibility. They aren't treated as subhuman or anything like that, but are definitely discriminated against based on their gender.
Isolation is a second theme of the book. The protagonist, Menolly, runs away from home. Her father, in keeping with the gender roles theme, told Menolly that she may no longer pursue music. Menolly runs away and survives by herself in a cave. She is not totally alone, if you count the baby dragons that she finds and decides to raise, but there is no human contact.
A third theme is choices and consequences. I don't want to make it seem like the consequences were bad, but Menolly's choices were points of no return for her. She chose to run away knowing she would not come back. She chose to raise the dragons, which led her to become extremely resourceful. Thinking of this theme reminds me of Frost's "The Road Not Taken." He knows that one thing will lead to another and he will not be able to return. Menolly knows this too when she makes her choice to run away.