When: Anytime it’s not too cold or raining. But I would recommend the summer when there is a chance to watch a play there.
Get there: There are trains and busses to get to the area. However I would definitely rent a car. It’s the easiest way and you can go wherever you want and explore the beaches and cities of Peloponnese.
Price: If you want to watch a play prices are more expensive (and also the plays are booked out pretty quick). If you just want to visit the archeological site it’s quite cheap. Also if you are a student you get reduced theater tickets and most museums in Greece are free or reduced for you.
Do: Try and go for the yearly theater festival. It’s so special sitting and watching a play knowing that people over 2000 years ago did exactly the same in that particular theater. There are sometimes also plays in english for those who don’t speak Greek.
My tip: Try the acoustics. If you stand in the middle of the stage and speak normally. Somebody on the top will be able to hear exactly what you’re saying. It’s really amazing considering the theater fits about 14000 people. A lot of plays today don’t use microphones for that reason.