Dishonesty happens in the form of:
1. The double lives of each character. Algy likes to escape to the country by saying he'll visit a (fake) friend named Bunbury, while Jack escapes to the city by visiting a (fake) brother named Earnest.
2. Earnest is also the fake name that Jack uses while he is in the city. Gwendolyn, the woman whom he wants to marry, happens to have a silly fixation with the name. When he proposed to her, he didn't fix the fact that his real name is Jack (John), and so he tries to get baptized but won't admit to the truth.
3. Equally, when Algy wants to go meet Cecily, he claimed to be the (fake) brother of Jack's whom Jack visits in the city (every time he wants to escape). Hence, she is also deceived by the fake name (and character) that Algy is pretending to be.
4. Jack and Algy are also dishonest with each other. Algy did not know about Jack's double life as Earnest in the country. He found that out through Jack's leaving a cigarette case inscribed for Earnest from "Little Cecily."
5. Algy is a Dandy, yet does not pay his bills. He is "terribly hard up", yet, he lives the life of a gentleman, keeps a servant, hangs out with the upper classes (which he is a part of), and still is elitist.
The first two lies are about the name Ernest. Jack claims that he is "Ernest" when he is in town, and Gwendolen and Algy believe that is his name. The fact that he proposes to her under the name of Ernest is a big lie.
Then Algy pretends that he is Jack's unfortunate brother Ernest to Cecily. She believes Algy is his brother and that his name is also Ernest.
Then Algy also continues to lie to his Aunt Augusta about having a sick friend Bunberry who needs him. He only has to visit this "fictional friend" when she invites him to family dinners and parties. He constantly lies to get out of spending time with his family.