In Romeo and Juliet, how do Romeo and Juliet react to each other's family affiliations?

When Romeo discovers, via the Nurse, that Juliet is a Capulet, he says that his "life is my foe's debt." He means, basically, that his bitter enemy has given him his sole reason to live. He is absolutely smitten with Juliet, and in this line, he acknowledges the complexities that will result from this fact. A few lines later, Juliet asks the Nurse to identify Romeo, saying that "if he be married, my grave is like to be my wedding bed." In other words, she says she will die unmarried if she cannot marry Romeo. When the Nurse reveals that Romeo is in fact a Montague, Juliet bemoans the fact that her "only love" is "sprung from [her] only hate!" Later, she reflects on the tragedy of this fact, asking rhetorically "What's in a name?" She says that if Romeo's name were any different, he would still be the same person--"a rose by any other name would smell as sweet"--and begs, in her soliloquy, for Romeo to "doff" his family name. She, like Romeo, realizes the implications of her love for a man who is denied to her by her birth. 

What are Romeo's and Juliet's reactions when they discover they love their family's enemy?

Both Romeo and Juliet learn the horrible news about each other's familial relationship sin Act 1, Scene 5. The nurse must be the bearer of bad news to each of them. She first warns Romeo by telling him he has set his sights on the daughter of the lady of the house. Romeo is flabbergasted. "Is she a Capulet?" he cries. "O dear account. My life is my foe's debt." Benvolio tries to dissuade his love-struck friend. "Away, be gone, the sport is at the best," he advises (1.5.115-20).

Juliet's nurse is then has the unpleasant task of telling her charge:

Nurse: His name is Romeo, and a Montague, / The only son of your great enemy.

Juliet: My only love sprung from my only hate. / To early seen unknown, and unknown too late. / Prodigious birth of love it is to me / That I must love a loathed enemy.

This is a high tension point because it is at this juncture that both of the young lovers must decide to break off on their own, to "deny their families, (and) to refuse (their) names." Doing so could lead to dire consequences. As children of the elite, they are dependant on the fortunes of their families, dowries, and connections to get along in the world. They have no trades of which to speak and nowhere to go. It is likely, given the history of the feuding families, that both would be disowned.

How do Romeo and Juliet feel about each other and their families?

The reader does not know much about Romeo's parents.  They are not really in the play.  Montague gets fewer than a dozen lines over the course of the entire play.  What the reader or viewer learns about Romeo's family is from Romeo himself.  The fact that Romeo spends more time with Friar Laurence and seeking his advice is an indicator that Romeo is not that close to his parents.  Another indicator of this is that Romeo is very willing to throw away his family name in order to be with Juliet.  He comes from a wealthy, respected family, so being that willing to give up the name means it must not mean much to him in the first place.  "Henceforth I shall never be Romeo...My name, dear saint, is hateful to myself / Because it is an enemy to thee" (2.2). 

Juliet's relationship with her parents is much better, at least at the start of the play.  Paris asks Capulet if he can marry Juliet.  Capulet's response is yes, but wait, and only if she agrees to it.  That's a fairly nice gesture since it is well within Capulet's power to simply tell Juliet who she will marry and when (which is what happens like two days later). Before the party, Juliet's mom asks Juliet to keep an open mind about Paris and talk to him at the party.  Juliet agrees.  She listens to her mom and agrees to follow directions.  Solid relationship.  

Of course all of that changes when she meets Romeo and falls in love. Then she doesn't listen to a word her parents say.  She marries behind their backs and tells her father that she will not marry Paris no matter what.  Capulet hits her and tells Juliet to leave his sight.  That's a quick turn around from the happy and content family a few days earlier.  

What are Romeo and Juliet's feelings towards their families?

In a nutshell, both of them are willing to throw away their families for their love for each other.  That's overly simplistic though, so I'll discuss each in turn.  

Romeo is probably not close to his family. Evidence of this is how little they are in the play.  Montague gets fewer than 12 lines over the course of the whole play.  The best way to glean information about Romeo's family is to study Romeo himself.  There are times throughout the play that Romeo seeks advice from an adult male. It's a perfect opportunity for him to consult his father, but he doesn't.  Instead he goes to Friar Lawrence.  I was serious when I said that Romeo was willing to cast off his family and family name for Juliet.  "Henceforth I shall never be Romeo...My name, dear saint, is hateful to myself / Because it is an enemy to thee" (2.2).  That's a big deal because Romeo comes from a respected, wealthy family.  To be willing to throw all of that away so casually indicates that Romeo doesn't care much for his family in the first place. 

Juliet's relationship with her parents is not as bleak as Romeo's family relationship. Juliet probably has a good relationship with her father, because it's clear that Capulet cares for Juliet. Early in the play, Paris asks to marry Juliet.  Capulet says yes, but that Paris has to wait since his daughter is only 13 years old.  He also tells Paris that it will only happen if Juliet agrees to it.  Capulet has the power and ability to force his daughter to marry whoever and whenever, so the fact that he wants Juliet to be okay with it shows that their relationship is solid. Juliet's attitude toward her mother is equally gracious.  Lady Capulet speaks to Juliet about Paris before the party and Juliet politely listens and agrees to keep an open mind about Paris as her mother suggested.  Again that indicates that Juliet has a healthy relationship with her mother.  It's not strained or antagonistic in any way.  

Of course Juliet's familial attitude takes a 180 degree shift after meeting and falling in love with Romeo.  She couldn't care less what they think then.  She marries Romeo behind their backs.  Flat out tells her mom and dad that she will not marry Paris, despite being required to do so by Capulet. Capulet hits her and threatens to throw her out. Whoa. That's a quick attitude shift.