Super Mario Run includes a few ways to add friends, but we've had the best luck adding friends using their IDs. (Adding friends through social and links don't always work.)
On the Kingdom Builder screen, tap on the Friends icon in the upper left side.
Your Friends List will open up and the various choices for connecting with friends will be at the bottom of the screen. Ignore everything but the "Add" icon:
Tap the Add icon and add friends by typing in their player ID. This is also the screen you would use to find your ID. Unfortunately, you'll need to ask your friends for their IDs since there's no way to search for IDs.
Tap on the Accept icon to open a screen where you can acceptfriend invitations.
You can also add friends using these options, but people have reported problems with them since the app launched. Proceed with caution:
Tap the Link icon. A screen will pop up that generates a link to your account that friends can use to join you. This link can be copied, emailed or sent through iMessage. Just tap on the icons at the bottom of this screen to share.
Tap on the Invite icon to invite friends using your Twitter or Facebook accounts. Just tap on the social media icon of your choice.
That's all there is to it. Right now adding friends only allows you to see their score. There are rumors that in the future you may be able to send your friends gifts or race them. These haven't been confirmed, though.