What are the main characters' motivations in The Impotance of being Earnest?
One could argue that Jack Worthing is the protagonist of The Importance of Being Earnest, but there are five central characters whose motivations are most important to the play.
Let's start with Jack. When we first meet Jack in act 1, he is visiting his friend Algernon in London. During their conversation, it is revealed that Jack, who has been going by the name Ernest in London, is named Jack and not Ernest; somehow Algernon never knew this before. We learn that Jack pretends to be "Ernest," who he claims is his brother (he is invented), while in town so he can get into "scrapes" without word of his bad behavior reaching his ward (Cecily) or other people who respect him in the country. However, Jack wants to reform and become Ernest so that he can marry Gwendolen Fairfax.
Algernon is more playful than Jack and is mostly motivated by pleasure and enjoyment. Algernon reveals that he has an invented friend named Bunbury, who he simply says is ill any time he (Algy) wants to avoid a social obligation. Algernon calls his adventures "Bunburying." By the end of act 1, Algy is determined to "Bunbury" to Jack's country estate to meet Cecily.
Gwendolen Fairfax and Cecily Cardew have the same goal: they want to marry a man named Ernest. This is part of Wilde's satire of the Victorian upper class's requirements for marriage. Members of this wealthy class based their choice of spouse (or their daughters' spouses) on last name, status, and property. One could argue these are superficial concerns and have nothing to do with personality or individuality. To poke fun at the importance of family name, Wilde has Cecily and Gwendolen obsess over a random first name. The name, of course, takes on further ironic significance because of the play on words between the name Ernest and the adjective "earnest" (sincere or genuine).
Finally, Lady Bracknell is a major character whose motivations are clear to the audience. She aims to marry her daughter, Gwendolen, to a suitable husband. She interrogates Jack in act 1 to determine whether he is appropriate and finds out that he does not know who his parents are. This is obviously a deal-breaker. Later, we also understand that Lady Bracknell is motivated by money, because she will allow her nephew Algy to marry Cecily only after learning that she has a large inheritance.
That's an interesting question, in that the motivations of everyone in this play are what push the plot forward. Not every play is like that; many have external events providing the main push for the action. But in this play, while there are silly coincidences and events which do move the plot along, the motivations of everyone are so clearly the point of the play that the events seem less important.
Jack Worthing wants more than one thing; he wants to go to town and be a gad-about, and generally get up to no good, but also to preserve his reputation and be a good example for his young ward Cecily. So, because he wants to have it both ways, he invents his fictitious brother Earnest to cover up his indiscretions.
Jack also wants to marry Gwendolyn Fairfax. Gwendolyn wants to marry Jack. Lady Bracknell, Gwendolyn's mother does not want Gwendolyn to marry Jack because he is "unsuitable". Lady Bracknell's motivations are to protect her daughter and to uphold convention.
Cecily wants to marry Algernon. Without getting too much into the complications of the play, Algernon wants to marry Cecily, but when he meets Cecily and falls in love with her he is posing as the (fictitious) brother of Jack, Earnest. You can imagine the hijinks which ensue, but the motivations of both Algernon and Cecily are to marry each other. Algernon has a secondary motivation which results in his fictitious friend Bunbury. He has created this imaginary friend so that he can get out of social obligations, especially avoiding his aunt Lady Bracknell.
The Reverend Chasuble wants to marry Miss Prism. Miss Prism wants to marry the Reverend Chasuble. There are some complications because Miss Prism has a slightly scandalous past (involving Lady Bracknell), but Miss Prism's and Rev. Chasuble's desires are essentially simple.
As you can see, Lady Bracknell is, essentially, the fly in the ointment for everyone. When her desires are resolved everyone else's are, too. Wilde plotted this comedy perfectly, with everyone's wants dependent on someone else's. When the snags are removed, everything comes out right in the end.