How do I start a game with a friend?
Tap on New Game, then choose Game Center at the top. Then tap on Match, and you'll see you a popup with recent players. If your friend is listed there, great, otherwise, tap on Invite Friends at the bottom, and it puts you into Messages, where you can send an invitation to your friend. They should get that invite, and can start the game by accepting the invite.
What’s the difference between Othello and Reversi?
This is well explained in this article by former World Champion Ben Seeley. In short, Othello is a trademarked name for the game as it’s played now, while Reversi (the name of an older, inferior game) is often misused as a name for the same game.
Is there a book about Othello that will help me improve?
Yes, there are two excellent books on Othello strategy, both available for free as PDFs. The first one is “Othello: Brief and Basic” by Ted Landau; you can download it directly from his website at http://www.tedlandau.com/history/othello.html. The second is “Othello: A Minute to Learn ... A Lifetime to Master” by Brian Rose, which you can download at http://www.ffothello.org/livres/othello-book-Brian-Rose.pdf.