Where does Elizabeth want John to go, and what does she want him to do there ?
I believe you're talking about Act II, Scene i where Elizabeth asks John to go to Salem and tell the court that the witchcraft accusations are a hoax. Abigail admitted to John that the girls were just playing around in the forest and lied to keep from getting into trouble. Mary Warren has told Elizabeth that fourteen people are now in jail and will be hanged if they don't confess. Elizabeth realizes that this has gone too far and could have terrible consequences if the proceedings aren't stopped immediately. John is reluctant to tell the court because he can't prove what Abigail told him because they were alone when she admitted there was no witchcraft involved. Elizabeth is even more upset to discover that Abigail and John have been alone, and she thinks John is hesitant to tell the court because he wants to protect Elizabeth.
You have not said what Act you are referring to, but in Act II... there is a scene which places Elizabeth and Proctor and Mary Warren talking about what has gone on in the court today. We find that Elizabeth's name has come up, although Mary Warren claims to have helped not see that accusation come to fruition.
Elizabeth wants John to go to Abigail to make sure they talk this out and that Abigail will not accuse Elizabeth.
She says this because
There is a promise made in any bed.
Knowing that John has slept with Abigail, he should have a power over her to make her act accordingly. Elizabeth fears that Abigail will have her killed.
Elizabeth wants John to go to Salem to let the judges know that Abigail is not being truthful and she told him that the girls were "playing" in the forest.
Abigail has accused Elizabeth in court at this point in Act II of The Crucible. Elizabeth knows that this is an attempt of Abigail to take revenge and to have John Proctor all to herself. So as to relinquish any further attempts Abigail might make to sabotage Elizabeth and John's marriage, Elizabeth begs John to go to Salem and speak to Abigail. She wants John to make it clear to Abigail that there is no hope or chance for them to be together, this way, Abigail will let go of John and forget about trying to get rid of Elizabeth.
Elizabeth also wants John to go to the court and explain to the court what Abigail told him in confidence in Act I. John does not want to go because "[Abigail] told it to [him] in a room alone."