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.