There are many themes in A Midsummer Night's Dream, and there are many clues that tell us the themes. In order to derive a deeper understanding of the themes, it helps to be able to pick out important occurrences and ask yourself what theme it could be portraying. For example, the story line shows us a great deal of contrast between the city of Athens and the forest. Everyday occurrences take place in Athens while Shakespeare creates a dreamlike existence in the woods. You can discover one central theme by asking yourself, what does this juxtaposition between everyday life and magic show us? The mechanicals also present the same theme; therefore, you can figure out this theme by asking yourself why is the illusion of the mechanicals' play so important? Why are their ambitions so important? Also, Puck presents another central theme, especially in his famous line, "Lord, what fools these mortals be!" (III.ii.116). We can figure out what theme Puck is representing by asking ourselves what things he is seeing that he thinks are foolish behaviors? Also, why is this comment important?
You can do something similar for symbolism. Shakespeare certainly does make use of some symbolism, which is primarily seen in recurring motifs. Nature is especially a recurring motif in the play. We frequently see references to the moon, the forest, and especially the magical flower. The moon is mentioned by Theseus who is waiting for the new moon in order to hold his wedding day with Hippolyta, as we see in his lines:
[F]our happy days bring in
Another moon; but, O, methinks, how slow
This old moon wanes! She lingers my desires. (I.i.2-4)
The forest is referred to as a place of escape, a place where Hermia and Helena bonded as children, and now as a place where their friendship is being broken apart and, yet, also as a place where the two couples unite. Two questions that can help you explore the deeper meaning behind the symbolism of this recurring imagery would be: What does both the full and the new moon symbolize? How does the forest symbolize man's dual nature?