A way to get history wrong is to apply a modern understanding or concept to a historic context. One such term is religion. The term itself is actually rather modern and deviates substantially from the classical or even medieval definition. As such, seeing the image of a classical church when you visit a Greek temple is misplaced.
Epidauros shows this quite nicely. Devoted to the Greek god of healing (Asklepios) this nominally is a temple, i.e. a religious institution by our modern standards. But in reality you are rather in a medical facility where patients would seek treatments for ailments and the priests acted as physicians. The treatments are very similar to what you find in a spa town nowadays.
The highlight of the site, by far, is the amphitheatre. Akin to modern spa towns with their theatres, casinos and concerts, culture and entertainment were considered part of the treatment. The scale and the state of preservation of the amphitheatre are exemplary. In addition, you find a stadium and plenty of ruins of old temples and medical facilities.
At some distance and apparently outside the fenced are of the main site is the Sanctuary of Apollo Maleatas. I missed it as I had assumed it to be within the premises or at least see some signs for it. I didn't and when I had realised it, I was already on my way back to Nafplio.
Overall, the site is less spectacular than e.g. Olympia or Delphi. It's a big bunch of ruins plus an amphitheatre. To me, it's the least interesting of Greek ruins.
Getting There
Modern (Nea) Epidaurus is on the coast, some 20km from the archaeological site up in the hills. During high season you should find a bus connecting you to Nafplio or Nea Epidauros. However, ... the schedules offered don't really work out well, even if they were running. When I visited, tourist levels were low, so the bus did not stop at the site. Best option is to book a return cab from either Nafplio or Epidauros.
In summer, they have theatre performances on site. There are dedicated connections then from Athens and extra busses from Nafplio.
While You Are There
Nafplio is a pleasant city and a good base to explore the area including Mycene.Argos also offers interesting ruins including another Greek amphitheatre (T).