Rules of Grammar

I'll just answer with a simple question. Why do you doubt that it is grammatically correct?

My friend and I went to the movies.....

That construction always looked awkard and clumsy to me. But as it seems, it's got to be that way, if you don't want to look like an uneducated, lower-class person.
What do you say if you take your five year old daughter for a trip - do you say may daughter and I are going to Zimbabwe, or my daughter and I are looking for bargain flights Madrid to Lima?

Tonieja, yes, that is what you say. Why wouldn't you? That is, assuming your five-year-old daughter actually was actually joining you in looking for bargain flights. That seems a bit unlikely to me, but I don't know your daughter.

OP: I is in the nominative case, the case we use for subjects of sentences. I went to the movies. We (nominative plural) went to the movies. My friend and I went to the movies. Me is the accusative/objective case, the case we use for objects of sentences. The movie didn't interest me. The movie didn't interest us (accusative plural). The movie didn't interest my friend and me.

Caballero: An easy way to remember whether to use I or me in examples like yours is to omit the "other" and see if it sounds right. You wouldn't say "me went to the movies," so it has to be "My friend and I." Likewise, you wouldn't say "The movie didn't interest I," so it has to be "...my friend or me."

To tonieja: I think that "My daughter and I are going to Zimbabwe" may seem odd if she is five years old and you think of that grammatical construction as involving equals: "My wife and I," or "My husband and I" might seem less strange. If that's what prevents you from hearing it as normal, you could rephrase the statement as "I'm taking my daughter to Zimbabwe."

I think we would all find your second example strange: "My daughter and I are looking for bargain flights" only makes sense if she is a precocious child showing you how to do on-line searches. Otherwise it's "I'm looking for bargain flights, because I want to take my daughter to Madrid."

Thanks VinnyD and NorthAmerican.
It's kind of obvious to me that between two equals - husband and wife, friends, or partners, it's 'my wife and I' type of construction.
However, in two different topics on SEA branch, that follow after each other, I read:
"Me and my fiance are planning a round trip to South-East Asia this winter"
and
"My partner and i are heading to Laos from north Thailand"

How does that sound to you? Is the form in the first example wrong or acceptable for a native English speaker. Or, is it something in class with "definately", that I see too often in posts from many native English speakers too?

"Me and my fiance" is wrong despite its being used by many thousands of native speakers of English. We are vaccinated against that construction in elementary school, but it has continued to infect speech and writing, and seems to be thriving.

tonieja, I'm intrigued by your question. What is it about "my daughter and I..." that sounds wrong to you? Is it that the two people in that phrase doesn't seem equal, and therefore it should be different from "my wife and I"? Or is it something else?

It's very interesting that that sounds wrong to you.

OP that's correct. "The movie didn't interest either my friend or me" sounds more natural , but you don't really need it.

Are you a Spanish speaker, by the way? Spanish would use the subject pronoun "yo" there, wouldn't it?

I'd be interested if anyone knows the reason for the difference between the languages in this case. It's something I've never quite understood.