Which phrase is more correct? (inside)

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.

It depends on whether you and your family is the subject or object of the sentence.

My family and I went to the seaside.
The rain fell on me and my family.

I don’t think you need to add ‘members’ to ‘family’

You should always say the other person before yourself also.

Agreed. ” My family members and I”.

Depends on the rest of the sentence.
“They invited my family members and me to a show,” is correct.
My family members and I went to the show,” is correct.
To determine whether to use “me” or “I”, imagine saying the sentence without including the family members. You would say, “They invited me to a show.” You wouldn’t say “They invited I to a show.”
Conversely you would say “I went to the show.” You would not say “Me went to the show,” (unless you’re about three years old!)
And @janbb is correct. In any situation involving multiple people you always put yourself last.

If Your family members and you are the subject of the sentence, then the proper phrasing is “I”. If you are the object of the sentence, then use “me.”

I.e., My family members and I did something.

Or They did something with (or to or for) my family members and me.

It depends chiefly upon whether the phrase is the SUBJECT or OBJECT of the sentence:

My family and I threw a party.

The neighbors threw a party and invited my family and me.

Wow. Does Fluther have a Hall of Redundancy Hall of Fame. Because I’m nominating this.

What? Do you think people are all saying the same thing?

If so, I think there’s some redundancy in that.

If it’s narration, you would say, “My family members and I…” Narration should always use proper grammar.

But if it’s dialog, you would say it however the character would say it. But your character must be consistent throughout (unless your character is slowly improving his grammar as the story progresses).

Yes, everyone is saying the same thing @wundayatta. I dropped all the preposition / object / subject stuff and just told the OP something simple.