If you're sitting an English test "A friend and I." In real life, either. Another thing I would say is "me and a friend" I don't have a clue if it's grammatically correct or not, though.

'A friend and I' is the grammatically correct form, however 'A friend and me' is very commonly used and you won't be misunderstood if you do happen to use it. 

'My friend and I' is the grammatically correct form. It's right, but it can sometimes sound overly formal. As Spangola says, this is the one you'd use in an exam.

'Me and my friend' is acceptable nowadays in in casual colloquial spoken language. It sounds natural in informal contexts, even though it's not 'correct' as such. Personally, I'm with Spangola on this one - this form is so widely used that it doesn't actually matter whether it conforms to conventional grammatical rules. Let's just call it 'non-standard', which is linguists' shorthand for anything that's part of the language but not strictly 'correct'.

'My friend and me' doesn't sound natural, however. Don't use this form - choose one of the two options above, depending on the context.

Definitely not "A friend and me ...". Think of it this way: if you drop the friend and keep it to a single person, would you say "Me am making a group"?

By the way, although "making" is perfectly correct syntactically, I think "forming" or "setting up" or something like that would be a more colloquial verb to use and would sound more natural.

Ok, A friend and I, thank you both of you.