Both are correct. But it depends on the sentence.
Use ‘my friends and I’ where you would use “I” (i.e. as the subject):
My friends and I went bungee jumping last weekend.
And, use ‘me and my friends’ where you would use “me” (i.e. as the object):
My little sister was upset that she wasn’t allowed to go bungee jumping with me and my friends last weekend.