At the party, Romeo is spotted by Tybalt, who is a enemy of the Montague family. This angers Tybalt because he believes Romeo is there "to fleer and scorn at our solemnity", which means Tybalt believes that Romeo is only at the party to mock the Capulet family. Because of this, Tybalt wants to "strike him dead". This is a man vs. man conflict because Tybalt wants to fight Romeo, but is not allowed to. This leads to Tybalt challenging Romeo to a duel, which eventually leads to Romeo's banishment.
After Romeo kills Tybalt because Tybalt killed Mercutio, he is quickly banished by Prince Escalus and filled with grief. Romeo, who doesn't know what to do, tries to stab himself. This is a man vs. self conflict because Romeo, who is banished from Verona, cannot meet Juliet anymore.
After Tybalt's death, Capulet decides to immediately marry Juliet off to Paris. However, Juliet refuses to go with the wedding, which leads to a heated argument between Capulet and Juliet. Capulet states "I tell thee what: get thee to church o' Thursday,/ Or never after look me in the face", showing Capulet will disown Juliet if she doesn't marry Paris. Therefore, this is a man vs. man conflict between Juliet and Paris.