The most important thing to remember when studying a play is that most plays were not written to be read but to be performed in front of an audience. It is important to pay attention not just to what is said by characters, but to whom, to who else on stage can overhear this and to how other characters would react. Also pay attention to stage directions and to implicit directions in what characters say so that you can form a mental picture of what characters are doing. Consider how characters' lines might be spoken differently by different actors and how this would affect the meaning.
A good exercise is to look at a play from the perspective of a director, a casting director, a props and scenery manager, a costume-team leader or a lighting engineer. Think about the decisions that all these different people would make in order to bring the script to life on the stage. It is very difficult to understand how plays work as theatre without having first-hand experiences of performances, so if it is at all possible, you should see a live performance of the play that you are studying. Failing this, watching film versions of the play is also helpful. Seeing more than one production is a good way to understand how different directors can interpret the same script in very different ways and will help you to start thinking about what you would do in their position.