Making Friends the Easy Way
If you click or tap on another player's avatar a "mini profile" opens with a number of options. One of those options is Add Friend. Select this option, and that player will get a pop-up asking if they accept your request. If they agree they're added to your friends list. If not, nothing happens. Some players have deactivated this feature, but you may still ask them to be friends the not-quite-as-easy way.
Click the Add Friend button in a player's mini-profile. If they accept, you'll each be added to the other's friends list instantly.
Making Friends the not-quite-as-Easy Way
You can also invite somebody to be your friend from their main profile page. You will find the Invite Friend button in the upper left area of that screen. If you ask them this way, they get an inbox message asking if they accept. This is how you can ask somebody who isn't in the room with you to be your friend.
You can also remove a friend by selecting the More option from a player's Profile, and then clicking Remove Friend.