Why do characters in The Crucible lie and what are the objectives of the ones who tell the truth?
There are a number of reasons for why characters lie in this play, but the simplest way to answer the question is to say that the lies serve their desires and protect their interests. Every character has a different set of desires and interests. One very prominent liar is Abigail Williams. Abigail lies about her activities in the woods because she doesn't want her uncle to think she is disobedient, particularly because she was recently fired from her job as a servant for the Proctors. Abigail lies about seeing various townswomen "with the devil" because she knows that showing she is caught up in the hysteria will help destabilize the town. Abigail believes she will get her way if the status quo is breached. Abigail wants John Proctor to leave his wife for her, and thinks this can happen if she lies about Elizabeth Proctor and impugns her good name. Abigail also lies in the courtroom, pretending to see things that support her narrative of bewitchment. All the lies Abigail tells are self-serving and aimed at getting what she wants.
As for characters who tell the truth, the main reason they do so is because they are honest by nature. The characters who make a point of telling the truth do so to counter the accusations against them, and to express their integrity and piety. Characters such as Rebecca Nurse and John Proctor are known to be honest, and they believe their good reputations are sufficient for people to believe they are telling the truth. They soon learn, however, that their words can be twisted in this atmosphere of hysteria. As they attempt to unravel the mystery of the witchcraft accusations, the judges are just as likely to believe lies as truth.