The friar devises a strategy: Romeo will pay Juliet a visit that night, but will depart her chamber and Verona before the dawn. He will then live in Mantua until the word of their marriage spreads. He must prepare to pay Juliet a visit before fleeing to Mantua.
Romeo visits Juliet at her father's house where she lives with her family. Her father is not home so they can talk freely. They discuss their love for each other and how their families cannot approve of this match because they are from different classes. However, Romeo does not care about these things and only wants to be with Juliet. He tells her that he has something important to tell her but first he needs her help. He asks her to go out into the garden and look for a rose without any thorns. When she finds it, he wants her to bring it back to him.
Juliet goes outside and searches for the rose but cannot find one that fits Romeo's description. She returns inside and tells Romeo that there are no roses in her garden. Romeo tells her not to worry about it and that he will go gather some for her. Before he leaves, he gives her a little box with "something inside" that she is to keep with her at all times. He tells her that if anything ever happens to him, then the box should be given to someone who will know what to do with it.
Hearing this, Romeo attempts suicide but is stopped by the nurse. The Friar tells Romeo to go to Juliet that night as planned, and then depart to Mantua before sunrise. If she wants to see him again, she must come out to the church porch at dawn.
Romeo goes to a physician for advice on how to save his life, but the doctor tells him that he has no choice other than to wait and see if he gets better or not. When Romeo doesn't die, the doctor tells him that if he goes to Montague's house while still alive, they will have to cut off his head to remove the poison from his body.
Romeo decides not to go through with it and waits until he hears that Juliet has gone to the funeral. He then leaves for Verona to tell her about her father. Upon arriving in Verona, he finds out that Juliet has been sick with a fever since the day of her father's death. She is now in a critical condition and may not live through the night.
Romeo realizes that there is nothing more he can do for her parents so he goes back home to Mantua to tell Bianca and Lady Montague that he cannot marry Juliet. Then he plans to go to France where King Louis XII has offered him a post in his court.
Friar Laurence advises Romeo in Act 3, Scene 3 of Romeo and Juliet to go to Mantua, a town about 40 miles south of Verona, after Prince Escalus exiles Romeo from Verona for murdering Tybalt. Mantua was under the protection of France at the time and so Romeo can find safety there.
Romeo leaves Verona on foot and travels south toward Mantua. When he arrives in Mantua City, he goes to a house where he knows that Paris, the brother of Tybalt, lives. Here Romeo finds Paris, who has just returned home from war, and they rejoice at seeing each other again. Then Romeo tells Paris that he is in love with Juliet and that she returns his love. Paris says that he will tell Juliet of Romeo's visit. He goes to tell her that Romeo loves her and wants to marry her. As he speaks with Juliet by the river, Paris is killed by a falling rock.
After hearing what has happened, Romeo decides not to stay in Mantua but rather to travel again until he dies. He calls upon Paris' tomb to say goodbye to him, and then continues on his journey.
Mantua was located near a small village named Canio.
This is irrelevant to Romeo, and he remains, expressing his love for Juliet until the nurse arrives, calling for Juliet. Juliet kisses Romeo goodbye and promises to send someone to him the next day to check whether he intends to follow through on his pledge to marry her. Romeo sets off to find the Friar.
Now it's time to think about what happens next.
It is against the law to leave Mantua as an exiled person. If Romeo remains patient and waits for a brief period of time, the letter from Friar Lawrence may be brought to him and reveal the truth about the entire situation, which would affect the course of the play.
Friar Laurence expects that when Juliet is "returned to life," Lord and Lady Capulet would be so happy that she is alive that they will be reasonable about her marriage and work to improve their friendship with the Montagues. He assures Juliet that he would send for Romeo so that when she awakens, she can return to Mantua with him.
This is what drives Lord Capulet to madness and causes the feud between his family and the Montagues to break out into violence again.
Now, this isn't exactly why the story takes place in the first place but it's important to understand the roots of the conflict because it informs how everyone reacts to events later on in the story.
Also, note that while Friar Laurence wants Juliet to go back to Mantua, he doesn't actually have any authority over Prince Escalus or the Duke. He is simply a priest so he can do whatever he wants within the boundaries of religion but since he has no power over them, he cannot force them to let Juliet leave.
As for why Friar Laurence wants to send for Romeo... Well, as I mentioned earlier, he believes that if Juliet wakes up and sees that Romeo is still alive, then she will be forced to love him again even though she had given her heart to another. He thinks that once she realizes that Romeo still loves her, she will wake up from her deathbed and they can go home together.