Upon entering the friend option, the first thing you'll see is friend request, which takes up the middle tab at the top.
As you can see, there are 8 requests here. Press the green button to agree to the request, and red to reject. Obviously, you'll have to have free slots in your friend list in order to accept the requests.
First tab brings you to your friend list. Red icon deletes them off your list. Note that you'll only gain 5 slots for every 7th account level, and it hits a cap at 50 slots. Therefore, friend slots is a limited and valuable resource in this game.
Each time a friend uses you, he'll gain 25 Arcana coin. Same goes to you whenever you uses a friend. If you're using a guest or vice versa, you'll gain only 10 Arcana coin instead.
Last tab brings you to the friend search tab. Your ID is shown on the very top, and you can copy the ID with the brown icon on the side. Type in an ID in the dark green text box beneath it and press the button beneath it (which will turn purple) to search the friend and make the request.
Light green button brings you to the play store where you're supposed to search for friends in the reviews/comments, but...... oh well.
Top right corner hexagon brings you to the chat where you can get more friends there and etc.