Scenes One through Three make up the first act. Basically, we watch Joan convince everybody that she's legit. She starts with Captain Robert de Baudricourt, moves on to Charles and the Archbishop, and ends with Dunois. One by one she convinces these guys that she is the lady with the plan. By the end of scene three she's got everything she needs to fulfill the tasks that her voices have given her.
In the second act Joan's good fortunes begin to erode. Scene Four shows Warwick and the Chaplain plotting against her. When they team up with the well-intentioned (but perhaps incredibly misguided) Bishop Cauchon, we know that Joan is in some serious trouble. Oh, but her friends will help her out right? Nope. In Scene Five, King Charles and company give her the cold shoulder, telling her that, if she gets captured, she's on her own.
The play builds to its final peak and resolution in the third act. Scene Six gives us a glimpse into Joan's trial. Cauchon and the Inquisitor do their best to convince Joan to say she was wrong about the whole voices thing. In the end, Joan goes to her death for her beliefs. Shaw ends the play with an epilogue. This dream sequence lets us know that Joan was eventually made a saint by the same organization (the Church) that burned her for a heretic over 400 years earlier. The play ends with Joan asking God when the world will be ready for His saints.