The initial driving force behind the trials are the girls who make the accusations. Immediately following the impetus created by Abigail's lies, Reverend Parris and Reverend Hale are the main proponents of the trials.
Abigail and her friends are motivated by a need to deflect attention from themselves. They have been caught dancing naked in the woods and performing illicit rites. To protect themselves from punishment, they begin to accuse others of sending their spirits out and other witchcraft.
The trials, for the girls, represent protection from punishment. Parris is similarly motivated. He expresses his fear of losing his position in town if Abigail and Betty should prove to be guilty of witchcraft. When Abigail creates an opportunity for Parris to righteously pursue "justice" through trials, Parris energetically takes up the cause.
Hale has a different motivation. He truly believes in the existence of witchcraft in Salem for a time and stakes his reputation on proving it. His interest is more academic than any other character's interest.
Once the trials are underway, Danforth becomes the greatest advocate for continuing to see the will of the majority and then the will of the court expressed through the machinations of the legal system.
His association with the trials is professional, firm, and ultimately unyielding.