What are some mistakes in Romeo and Juliet that took the tragedy forward?

Romeo and Juliet's infatuation with one another is, in itself, not a mistake. The fact that they pursue their relationship in spite of knowing about the age-old feud between their families is where they err. Their relationship is doomed from the outset, but they naively venture forth.

Their statements about a name not meaning much is a foolhardy attempt by both to dispel the fact that in their situation, a name actually means everything. There is no way in which they are going to get around it except if their stubborn and vindictive families are to kiss and make up. This, however, is not about to happen anytime soon, as illustrated by the vehement encounters we read about. Even the Prince of Verona's threat of harsh punishment does not stop the families from fighting.

Furthermore, although Friar Laurence considers his idea of secretly marrying the two desperate lovers as an excellent method to bring an end to the feud, it is naive and foolish. In fact, such a step will complicate matters even further as, indeed, it does.

What worsens an already complicated situation is Romeo's noble refusal not to fight Juliet's cousin, Tybalt, because he now sees him as family. His rejection of Tybalt's provocation so angers his best friend, Mercutio, who sees Romeo's response as cowardly, that he takes up the challenge. His impetuous response, unfortunately, leads to his death when Romeo, in an attempt to intervene, gives Tybalt the opportunity to strike a mortal blow. 

Romeo decides to avenge his companion's death and kills Tybalt. This response leads to his banishment at the risk of execution if he should be seen in Verona.

The good-natured friar once again gets involved and arranges another of his well-meaning but ill-conceived plans. He provides Juliet with a sleeping potion which will put her in a deep coma for forty-two hours. Her family will assume that she has died and will bury her in the family vault. This action will prevent the ignominy of her committing bigamy by marrying the County Paris since she has already married Romeo. It will also give Romeo the opportunity to return to Verona and claim his bride, albeit from a burial chamber.  

Unfortunately, things do not work out quite the way Friar Laurence plans. Because of the plague, Romeo does not receive the desperate message, and the unfortunate young Montague learns from another that his beloved has died. The grief-stricken youngster buys a potent poison intent on sharing the life hereafter with his true love. The rest is tragic history.