To start an online match against a friend or random opponent:
From the main menu screen select the "Play" button. On the next screen select "New Game". The following screen shows your Facebook and/or Game Center friends who currently have the app, if any. Tap on a friend to start a new match with them.
On this screen you can also use the "Invite Friends" and "Add New Friends" buttons to invite friends who don't already have the game. Or use the Auto Match button to play a random opponent.
To play a local multiplayer game (single device, pass and play):
From the main menu screen select the "Play" button. On the next screen select "Single Device". This will start a single device game designed to be passed back and forth between players. You can only have 1 local game at a time. To reset your current game, swipe left on the Single Device option on the Games list.