Theatre of Dionysus
To get the best view of the Theatre of Dionysus you need to climb the Acropolis and look over the Southern Wall. The theatre was build in the sixth centure BCE, and holds about 15000 people. It is widely considered to be the first ever build theatre, and is the birthplace of Greek theatre.
PS: your Acropolis ticket includes the theatre, and you are welcome to wonder around the ancient site.