It depends on whether the phrase is the subject or predicate noun in a sentence, or an object.
Explanation:
1. As the Subject:
My family and I appreciate your kindness.
2. As a Predicate Noun / Nominative:
It was my family and I who arrived on time.
3. As a Direct Object:
They named my family and me as the winners.
4. As an Indirect Object:
Please give my family and me the opportunity.
5. As the Object of a Preposition:
The party was for my family and me.
N.B: The verb "to be" should take a predicate noun or adjective, not an object, but that's not the current colloquial usage. Most people say, "It's me," or "It's us / them," instead of the grammatically correct way: "It is I / he / she / we / they."