One of the appealing features of Apple Music is the social aspect, where you can curate your own playlists and share with others once you've put together one that's just right. In order to do this, you need to make sure you have your username set up. Here's how to do so:
First, launch the Music app (the new one with the white background and gradient icon) and take a look at the front page. Tap the person silhouette in the top left:
OK, let's get down to it boppers!
On the account screen, you'll see your name, and a space for an image and a nickname. Tap that area:
Tap the area with your name.
Also on a screen titled "Account" (because that's not confusing), there's an Edit button:
Apple Music account information. Again.
Now add whatever you'd like your nickname to be, and even adjust your name if you want to:
Well now you know my nickname on Apple Music.
For bonus points, tap the edit just below the circle with your initials in it and you can add an image from your device as your user image:
Just in case you weren't sure it was me.
Right now this nickname doesn't appear to do much aside from show up on your comments if you comment on Connect pages, but it stands to reason that in the future you can follow others (aside from artists with Connect pages) and find playlists by your friends.