To start a new game you must press the "Play" button, located in the center of the main screen, under your pawn, or in the new game space, located at the end of your list of pending games.
Once you do so, you will have to choose the desired game mode and, in case you select "Mountain Track" or "Toad Kingdom", you will be redirected to the "Select Opponent" screen where you can choose a random opponent or a Facebook friend (in case your account is linked to a Facebook profile).
Finally, press "Play" and you're ready to start the game!
Please note that in order to start a new game or play your turn in an existing game, you must have a balance of lives. You can see your life balance at the top of the main screen, to the right of your profile picture.