Since it is a portion of dialogue or monologue from a story, it’s correctness is dependent on the character. Is the character one who should be speaking in grammatically correct English? In which case, use ‘My family members and I…’ If the character is from a region where the dialect, coloquialisms, and so forth would suggest they know some proper English grammer and simply don’t use it, the first, or if the character has low education, then the second. It could also be written, ‘Me, and my family members…’, with the comma indicating a pause in speech and the next phrase being a thought that the speaker tacks onto the conversation.