(Since so little actually happens in this play, here's the very short list of things Edmund actually does. If you want a timeline of how his character develops through conversation, which is more important to a play like this, head to the "Detailed Summary.")
Edmund talks with Jamie in the dining room.
He comes into the living room and chats with the family.
When James and Jamie go to do yard-work, he lets Mary go upstairs to "nap" and joins the boys outside.
Edmund comes back in before the other Tyrones and does some reading.
When lunch is about to start, Edmund makes to swipe some bourbon, but he's interrupted by Jamie (who helps him steal it anyway).
Edmund and Jamie both talk with Mary before James shows up for lunch.
After lunch, Edmund goes upstairs after Mary because he has to get dressed for his appointment with Doctor Hardy.
He comes back down just as his parents finish talking about his birth, and leaves for town with his father and brother.
After his appointment, Edmund has a few drinks and comes home with his father before dinner.
Before his father returns from the whiskey cellar, Edmund can't take being with Mary anymore and leaves the house again before dinner.
He goes for a walk on the beach in the fog and stops at an inn for some drinks.
Edmund comes home around midnight and spends the remainder of the play talking to James, Jamie, and an oblivious Mary in succession.