Simply put, I think the city represents the real world of pragmatism, and the forest a dream world of escapism and transformation.
What we see of Athens, the city, in the first scene is a pretty uncommunicative marriage, in which Theseus won Hippolyta with his sword, followed by the insistence on an extremely harsh law which prevents Hermia from following her heart and marrying Lysander, instead threatening her with death.
Then, once the play moves into the woods, we meet the fairies -adding a whole magical dimension to the play. Hippolyta and Theseus' suppressed unhappiness transforms into the straight-out anger of Oberon and Titania.
The forest transforms people: it transforms, courtesy of Puck's flower-squeezing with the "love-in-idleness" juice, Lysander's affections for Hermia into love for Helena, and then Demetrius' love for Hermia into love for Helena as well. Moreover, when the mechanicals (who are acting - another mode of transformation) meet up to rehearse in the forest, Bottom is transformed into a half-donkey.
Translation and transformation are everywhere. And then, when the mortals return to the city, they take back some of the goodwill and happiness they learnt in the forest. Theseus exempts Hermia from the law he insisted on in the first scene, the mechanicals perform their play, and the lovers get happily married.
Hope it helps!