Me or I?

When speaking pretty much everyone uses 'me' but there IS a right and wrong way to refer to oneself.

Should it be 'my friend and I' or 'my friend and me'?: to decide simply remove 'your friend' from the sentence and go with the version that still makes sense.

For example:

My friend and I went shopping (correct: I went shopping vs me went shopping)

She gave it to my friend and me (correct: she gave it to me vs she gave it to I)