Is it "you and your family is" or "you and your family are"?

You should use are because "you and your family" is a plural subject.

When the subject of a sentence has two parts joined by "and" it makes the subject plural, so you should use a plural verb. Below are more examples of sentence subjects with two parts.

• Elizabeth and I are going for a walk.
• You and your dad win a lot of races.
• Cheryl and Sue are great friends.
• A book and a sandwich are sitting on the table.
• Sam and Kevin like to collect rocks.
• Josh and Chris listen to the same kind of music.