Character Clues
Character Analysis
Direct Characterization
Although there's not a narrator per se, the stage directions give us a lot of information and background about the characters. This is particularly useful for characters like Sarah, who don't get a whole heck of a lot to do in the play. Despite the fact that she doesn't get a lot to say, we know that Sarah is "slender, bored, and mundane" from Shaw's stage directions. Quite a few of the other characters (such as Lady Brit, Undershaft, and Stephen) also get some character background in this way.
Speech and Dialogue
The characters in Major Barbara are happy to give us their opinions on one another, which means we get a lot of characterization via speech and dialogue. Since Lady Brit is never shy with her opinions, she's a good place to go for examples.
We get a pretty decent idea of what to expect from Charles early on when Lady B offers her opinion of her future son-in-law during a powwow with Stephen about the family finances. In response to Stephen's suggestion that Charles might be able to increase his earnings by working hard until he comes into his inheritance, Lady B scoffs:
Charles Lomax's exertions are much more likely to decrease his income than to increase it. (1.31)
Later, Lady B makes clear that she disapproves of the way Charles talks and acts when she asks Dolly to "translate" for the other young man:
Adolphus: you are a professor of Greek. Can you translate Charles Lomax's remarks into reputable English for us? (1.114)
Incidentally, this instance of speech and dialogue also tell us a lot about Lady B and the strength of her sting . . .
Actions
Naturally, what the characters do also tells us a lot about the people they are. Shaw's stage directions have already told us that Andrew Undershaft is good-natured and easygoing, and that impression is confirmed when we see that the arms dealer is super laid back (and yet sweetly apologetic) when he forgets how many kids he has . . . and is unable to figure out who is who:
LADY BRITOMART: You look a good deal older.
UNDERSHAFT [apologetically]: I am somewhat older. [Taking her hand with a touch of courtship] Time has stood still with you.
LADY BRITOMART [throwing away his hand]: Rubbish! This is your family.
UNDERSHAFT [surprised]: Is it so large? I am sorry to say my memory is failing very badly in some things. [He offers his hand with paternal kindness to Lomax].
LOMAX [jerkily shaking his hand]: Ahdedoo.
UNDERSHAFT. I can see you are my eldest. I am very glad to meet you again, my boy.
LOMAX [remonstrating]: No but look here don't you know—[Overcome] Oh I say!
LADY BRITOMART [recovering from momentary speechlessness]: Andrew: do you mean to say that you don't remember how many children you have?
UNDERSHAFT: Well, I am afraid I—. They have grown so much—er. Am I making any ridiculous mistake? I may as well confess: I recollect only one son. But so many things have happened since, of course—er—
LADY BRITOMART [decisively]: Andrew: you are talking nonsense. Of course you have only one son. (1.148-157)
He stays cool and sweet even when Lady B is giving him what-for for not knowing his own kids, painting a picture of Andrew as a very good-natured—if not particularly family-oriented—dude.
Occupation
At first, the consensus among Lady Brit and her brood is that Andrew Undershaft's profession (and he, by extension) is pretty immoral. In Lady B's view, his profession is both how you know he's immoral and also a means of facilitating his immorality:
It is not only the cannons, but the war loans that Lazarus arranges under cover of giving credit for the cannons. You know, Stephen, it's perfectly scandalous. Those two men, Andrew Undershaft and Lazarus, positively have Europe under their thumbs. That is why your father is able to behave as he does. He is above the law. Do you think Bismarck or Gladstone or Disraeli could have openly defied every social and moral obligation all their lives as your father has? They simply wouldn't have dared. I asked Gladstone to take it up. I asked The Times to take it up. I asked the Lord Chamberlain to take it up. But it was just like asking them to declare war on the Sultan. They wouldn't. They said they couldn't touch him. I believe they were afraid. (1.41)
Of course, Undershaft manages to defend himself and his line of work so thoroughly that, by the end, most everyone in the family ends up thinking he and his profession are great—but whatever; for Lady B, at least, Andrew's profession is a big indicator of his fundamental immorality.