While "I and my family" is not technically ungrammatical, "my family and I" is the standard and polite form, placing yourself last in the list.

“John and I are going to eat fish pie at home this evening.” Using “I” is perfectly correct here, as John and I are the people who “are going to eat” – in other words, we are the subjects of the sentence. Almost everyone agrees that “I and John” however sounds odd, and thus no one uses this word order. 

It would not, however, be correct to say, “John and me are going to eat fish pie at home this evening”.