1. Hate destroys good things. At the beginning of Act Two, Scene Two, Friar Lawrence makes an observation about poisons: some plants can be both beneficial and poisonous, depending on how they are applied. This serves as a symbol for the power of love and hatred in the play and how they can come from the same source.
Romeo and Juliet's romance is something that ended badly, but not because Romeo and Juliet were "just dumb teens" as some would claim. Romeo and Juliet's love would not have resulted in death were it not for the feud. As members of noble families, they would have been well-suited for one another socially. However, the feud poisoned their romance and made it their demise.
The feud poisoned their relationship and resulted in their deaths. Hate also destroys Juliet's good relationship with her parents and results in Romeo's banishment from Verona. Only with the death of the lovers does Verona at last achieve peace.
2. Rashness leads to misery 95% of the time. A great deal of the problems in Romeo and Juliet emerge from impulsive behavior. Tybalt's slaying Mercutio and then Romeo's subsequent slaying Tybalt in a fit of rage sets the couple's final misfortunes into motion. Romeo's sudden decision to kill himself is entirely born of rashness and impatience. Even the adults are guilty of this: Lord Capulet's decision to have Juliet marry right away after Tybalt's funeral only pushed his daughter further away from him.
More patience would have yielded less tragic results to say the least.
3. Children and parents must have open communication between each other. Though many of the characters are teenagers, we only ever see Juliet interact with her parents. Romeo's parents are concerned about him but they cannot reach him, leaving Benvolio to figure out why he is sad at the beginning of the play.
Juliet has a mostly positive relationship with her parents at first. Her mother is a little distant and cold, but overall, Juliet's parents want her to be happy. However, when she grows up a little and goes against their wishes, they react violently, threatening to disown her if she does not marry Count Paris. They do not understand her sudden willfulness and she is frightened of telling them about her secret wedding to Romeo.
So much tragedy comes from a failure of communication in general in the play, but these failed familial bonds often prove the origin of those issues.