If you're staying in Athens then the easiest way to visit Epidaurus is by coach tour. These usually include a visit to Mycenae as well. As with most coach tours I find there isn't quite enough time to look at all the sights. But if it's all you can manage to fit in, then it's still well worth the visit.
You can get a bus from Athens to Nauplion, and then to Epidauros, but it takes quite a bit of time. If you don't use a coach tour company, then think about hiring a car. You might decide to make a day of it, and there's a choice of restaurants in the town of Ligourio which is just a short drive away.
When traveling by car my preferred method is to stay in Nauplion and drive out to the site. Nauplion is a lovely city and well worth a visit. It's a place you could stay for a few days anyway, and make it your base to visit other sites such as Mycenae and Corinth. Just down the coast from Nauplion is Tolon (Tolo), which while more of a tourist resort is still a good place to spend a few days or a week. You are also not far from the coast to the east, with access to the Saronic islands.
Map of the area
This map shows you where the site is in relation to Nauplion (just hover your mouse over the red markers). You can use the controls on the left to zoom in and out of the map, and you can move the map around by holding down your mouse button and moving the mouse around. If you select the "Satellite" view and zoom in to the site (labeled Asklipiio) you can clearly see the theatre.