The second act of Wilde's play The Importance of Being Earnest presents the setting and people from Jack's life in the country. Everything that was discussed between Algy and Jack in the first act are now put into play and tested. For example, Algy disagreed with Jack in the first act that married life was the optimal way to live one's life. In fact, Algy vowed that the life of a bachelor was the only life he would pursue; but then he meets Cecily and he all of a sudden changes his mind! As far as Algy is concerned, his conflict is definitely with himself and not necessarily other people. He disagreed with marriage, but changes his mind; that would lead one to determine that the conflict there is man vs. himself.
On the other hand, Jack has a different conflict--his future mother in law! Jack's conflict is certainly man vs. [wo]man because he has to convince her of his honorable lineage before he can obtain permission to marry Gwendolen. Jack is not fighting an inner conflict within himself; therefore, Jack's conflict is not man vs. self. In fact, Jack must also fight his past--something of which he has no pieces of memory to his personal history's puzzle. That puzzle is only solved when past secrets come to light.
The other conflict that may be pointed out in Act 2 is the one that encompasses the battle of the sexes. In this case, where the men are caught lying to the women about their identities, the conflict is most certainly man vs. [wo]man. The battle is declared open when Gwendolen announces, "I am afraid it is quite clear, Cecily, that neither of us is engaged to be married to anyone." The women then walk away from the men in perfect lover's-quarrel fashion and the war is on! It is then up to the men to decide if they will go after the women and fight for them, or to give up and surrender.
Craftily, Wilde choreographs all three conflicts to converge on stage at one time, creating one of the best comedies of the twentieth century to ever grace a stage.