How do Romeo and Juliet get married?

When Romeo sneaks into Juliet's garden, the two converse and share lovers' vows. Near the end of their conversation, Juliet says to Romeo,

If that thy bent of love be honorable,
Thy purpose marriage, send me word tomorrow,
By one that I'll procure to come to thee,
Where and what time thou wilt perform the rite (2.2.150-153).

In other words, Juliet plans to send her Nurse to visit Romeo to learn if he wants to marry her and when and where the ceremony will take place. Romeo then goes to visit Friar Lawrence, both to acquaint him with the news of Romeo's love for Juliet as well as to ask the friar to perform the wedding. Friar Lawrence agrees to marry the young couple because he hopes that

this alliance may so happy prove
To turn your households' rancor to pure love (2.3.98-99).

He wants to believe that the marriage between Romeo and Juliet will end the bloody feud between their families. Later, the Nurse visits Romeo, and Romeo tells her,

Bid [Juliet] devise
Some means to come to shrift this afternoon,
And there she shall at Friar Lawrence' cell
Be shrived and married (2.4.183-186).

He wants Juliet to tell her parents that she's going to confession, which would explain why she's going to Friar Lawrence's cell, and that's when they'll get married. We do not actually get to witness the wedding of Romeo and Juliet, but it would occur immediately following act 2, scene 6. At the end of the scene, the friar and Romeo and Juliet are just about to begin the ceremony.

Romeo and Juliet are married by Friar Lawrence just one short afternoon after meeting.  They wed in secret, with the bulk of the major players in the story -- Benvolio, Mercutio, and Tybalt among them -- completely unaware that the wedding has taken place.  The wedding is primarily arranged by Juliet's Nurse, who serves as a go-between between the couple.

While Romeo and Juliet are among the most famous couples in all of literary history, we don't actually see or hear any wedding vows in the course of the play.  The scene itself (Act 2, Scene 6) is only 37 lines long, as compared with other scenes that run into the hundreds of lines.  What we do hear are warnings from the Friar that moderation is key, and that Romeo and Juliet should not plunge headlong and passionately into their romance, as the results could be explosive.  Unfortunately, Romeo and Juliet do not heed the Friar's warning.

How does Romeo arrange his marriage in Romeo and Juliet?

In Act II, Scene 3, which immediately follows the famous balcony scene where Romeo and Juliet pledge their love for each other, Romeo goes to see Friar Lawrence with the express purpose of declaring his love for the "fair daughter of rich Capulet" and asking the friar to perform the marriage between he and Juliet. He hopes that he can convince the friar that he is truly in love with Juliet even though the day before he had been madly in love with another girl named Rosaline. Friar Lawrence quickly rebukes Romeo, condemning his idea of love:

Holy Saint Francis, what a change is here!
Is Rosaline, that thou didst love so dear,
So soon forsaken? Young men’s love then lies
Not truly in their hearts, but in their eyes.

Bid her devise
Some means to come to shrift this afternoon,
And there she shall at Friar Lawrence’ cell
Be shrived and married.