For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Act I
When you think of the first "Act" of this play, think of traditional familial-gender-racial roles as a bunch of bowling pins. Caryl Churchill sets them up for you and then gleefully knocks them down one by one. Oh hey, Betty is a submissive wife? Turns out she's cheating on her husband. Clive is a responsible husband? Well turns out he's cheating too. And that model son Edward? He's actually gay and having sexual relations with a pedophile named Harry. Oh, and the submissive African servant Joshua, who claims he "wishes he was white"? He ends up attempting to kill Clive.
Act II
Once Churchill has challenged these traditional roles, she doesn't just end the action there. She actually plays out what daily life is like when all of these roles go topsy-turvy. Churchill ain't being naïve here. She's not arguing that challenging roles is going to make the world into a utopian paradise. People are still going to be jerks to one another, as is the case with the way Gerry treats his lover, the grownup version of Edward from Act 1.
What Churchill is getting at in Act II is that people should always question traditional roles, because these roles result in oppression. Edward is scorned by his father (in Act 1) and his lover (in Act 2) because he acts in a traditionally feminine manner. Even though it's relatively socially acceptable for Edward to be in a gay relationship by the late-20th century setting of Act 2, his embrace of traditionally feminine qualities gets him ridiculed.
Act III
Once we've established the fact that traditional roles are finally getting questioned in Act 2, we return to the struggle of Betty. Betty is a loving traditional mother (think 1950s housewife), but she's played by a man in Act 1 and a woman in Act 2.
In a lot of ways, the common thread running through this play is Betty's struggle to stop defining herself based on what men want, and learning to enjoy life on her own terms. It isn't easy for her, though, since she's prone to anxiety and has a lot of trouble feeling comfortable without a man around to protect her from the scary world.
But ultimately Betty learns to accept herself. That's not to say that everything's dandy in her life. She's still lonely and she wishes that she could live with her daughter Victoria and son Edward. She still finds her children's fluid sexuality a little hard to swallow. But she's learned to embrace herself as an independent (and sexual) being, and for Churchill that's the first step on the road to abolishing oppressive roles once and for all.