“My friend(s) and I” or “me and my friend(s)” ?

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.