Shakespeare's plays can be divided into two broad categories: comedies or tragedies. Comedies have happy endings; in the most general terms, this means that no central character dies. Tragedies have unhappy endings; this means that major characters die.
Using this simple rule of thumb, that in a tragedy at least one major character dies, Romeo and Juliet is a tragedy. The fact that Romeo, Juliet, Mercutio, Tybalt, and Paris all die adds up to a tragically high death count.
Another characteristic that makes this a tragedy is that none of these characters needed to die; if there had been no senseless feud between the Capulet and the Montague families, all of these characters would have lived. Mercutio and Tybalt would not have entered into a sword fight if there was no feud. Paris would not have thought Romeo was planning to defile Juliet's grave if he did not think of Romeo as an enemy to the Capulets, and he would not have challenged Romeo to the sword fight that led to his death. Finally, Romeo and Juliet could have loved openly if there was no feud and avoided the mishaps that caused each of them to commit suicide.
A good way to approach the assignment is to create a file in your word processor listing the elements of Shakespearean tragedy from your class notes. As you read through the play, look for each of the elements, and then note down examples and quotations as you encounter them.
First, you will note that the play starts out in a temporarily stable but inherently precarious situation. The feud between the Capulet and Montague families is harming the city of Verona, but the situation is essentially a stalemate. When Romeo decides to attend the Capulet party and falls in love with Juliet, the situation becomes radically unstable.
Romeo and Juliet count as tragic heroes because they are noble and potentially admirable characters whose downfalls occur because they fall in love with each other and act on it. The audience is sympathetic to them. Their impulsiveness is a tragic flaw. The key conflicts are between the families and between the young couple and their families. The sad ending includes the death of both protagonists.