Translation:Me and my friend are cooking mushroom soup.
Nem összetett szó a gombaleves?
(And in English, "Me" shouldn't be used for the subject of a sentence.)
"I and my friend" is actually correct but it's considered rude (though grammatically fine) to list oneself first.
The English translation should be, "My friend and I are cooking mushroom soup." (By the way, this is one of my favourite soups my family makes, along with a similar one with cauliflower.)