We learn about how even royalty has to follow the law. So no one is above the law unless you are like Joe and have diplomatic immunity. By seeing that Princess Mia is being held up to a the law when it comes to her being married before ruling, we see that even though she rules the country she has to abide by the laws.
It also shows that sometimes people can think they are doing the right thing but they aren't. This also kind of goes with the topic last week of people using you to get what they want. Nicholas's uncle had him convinced that getting the throne was the right thing to do for various reasons. But in reality Nicholas's uncle was using Nicholas for his own greed.
We learn that a small act of kindness can mean the world to someone. Especially when it comes from a princess. Mia stops the parade in order to confront two little boys who were picking a on a little girl. She also showed compassion towards the children at the shelter, most of which were orphans. Mia bought them all crowns and had them be "princesses and princes" for the day and walk in the parade with her.
We also learn that sometimes we can lose it and that things will work out if you do. Even Mia has those moments. It just so happened that she was in the spotlight more than us. When she tells the queen, "sometimes I just lose it" we see the queen doesn't react well to it but in the end it worked out.