You can add and view your friends by going to the friends menu.
Share your Player ID so you can be added as someone's friend. You can also remove friends by going to the remove tab.
Tapping on the status icon will let you challenge them to a battle.
There are two modes available: Current Levels, which lets both players fight with the card levels and base health as-is, or Equal Levels, which sets both players' cards to level 1 and their bases to 10 HP.