Use "my friend and I" if you make up the subject of the sentence. It's not formal; this is simply standard English. Use "me and my friend" if you are an object in the sentence. You will hear people use "me and my friend" as the subject, but really, don't do it. Just speak normal English, and avoid copying our bad habits. No problem then. :)

Interestingly, I just had this discussion in my linguistics class. While we may be taught in school that proper English is "my friend and I," hardly anyone actually says that when talking. In conversation I wouldn't worry about it because native English speakers say "me and my friend" without thinking about it--although people will know what you mean either way you say it. In formal writing, though, I would use "My friend and I," since you're part of a compound subject and that's the right pronoun. Writing is probably the only place where people would notice or say something.

"My friend and I" is formal but in informal speaking you can say"me my friend/me and my friend"