The first one is correct. The reason being that the word, "family" is a singular noun and the verb has to agree with the subject. Even if the family consists of 15 or more people, the word itself is singular. I hope this helps.
ARE is used to conjoin plurals, such as: Our families ARE going to the US tomorrow.
I don't know how to explain it better, I established good grammar at a very young age, but don't remember all of the proper explanations for all of the rules.
"My family is" is correct, as "family" is a singular noun (even though it refers to a group of people). That said, I hear a lot of people saying "my family are" in informal, spoken English.