To add someone as a friend, simply tap on the player while in the lobby and select "Send Friend Request". This will send a friend request, which the player can Accept or Deny.
You can also add someone as a friend after a match is over. At the end screen you will be able to see all players and send them friend requests by tapping on the "+" button at the top.
You can see who is on your Friends List on the right side of the game's main screen. Your friends list is split into Online and Offline, so you know who is currently available to play.
To remove a friend, just go to your friends list and tap on the name of the player you wish to remove from your list.