To invite friends to the game, you must first select the Play option, on the main screen. There, you will find Mountain Track and Toad Kingdom, which are multiplayer game modes. If you select either of these, when prompted to choose an opponent, you will see the Invite friends option. If you select it, you will be offered different social applications through which you can send the invitation to your friends.
Once your friend receives the message and downloads the application, they will appear as an available opponent every time you choose to start a new game.