What details in Act I of Pygmalion suggest conflicts that might follow?
Some conflicts that are foretold in Act I of Pygmalion are foreshadowed in Liza's hysterical encounter with Higgins, The Note Taker; her encounter with Pickering; her encounter with Freddy, Higgins encounter with Pickering; and Clara's encounter with ... herself. Liza and Higgins begin on a high pitched note and they stay there throughout the play. Higgins offers Liza a chance for a transformation of their relationship to one of "fellowship" but she doesn't believe him and their conflict persists throughout the Sequel.
Liza's encounter with Pickering of a different sort. She speaks civilly, not hysterically, with him and he responds in kind with returned civility. He then responds with courtesy and spare change when she asks him to buy a flower and he cannot. Contrast this to Higgins who responds with discourtesy but tosses her a fistful of considerable money, enough money for her to treat herself to two taxi rides and language lessons (or so she thought ...).
Liza's encounter with Freddy shows a conflict of a different kind. she tells his mother that she called him "Freddy" just like anyone would do wanted to speak pleasantly to a stranger. When Freddy falls in love with Liza, his love isn't fervently returned, which presents a conflict of a different sort for Liza and for Freddy. However, Liza marries him anyway, which is the fulfillment of her "pleasant" feelings for him foretold in Act I.
Higgins' and Pickering's encounter establishes from the very start that Higgins is a bachelor and will remain a bachelor because he wants to be a bachelor and because he won't change his nature or his manners for anyone. This introduces the central conflict between Liza and Higgins: She may not be romantically in love with Higgins--or she may have taught herself not to be (there is some ambiguity in her behavior in Act V)--but she wants to be treated like a lady and with kindness. Higgins counters with the idea that he may not treat her the way Pickering does, but the real question is whether he treats anyone better than he treats her.
And Clara--well--the Sequel makes it clear that all throughout the time period of the play, Clara has been in conflict with most people and with most elements of society. Her mother could not manage to buy her an education, as the Sequel says, and so she is not intellectually or culturally at one with the social groups she thinks she ought to belong to. One day, she chances upon H.G. Wells, has the good fortune to meet him, and has her own life transformed (more of Shaw's belief in Life force) and new avenues of possibility opened up to her.