Why do I only see the live scoring page?

If you are on the iPad the navigation features are hidden when the iPad is oriented in portrait mode (taller than it is wide). As a result of this it's possible to launch the app and only see the live scoring page. To switch from this page rotate the iPad into landscape mode (wider than it is tall). You will see the navigation view appear on the left-hand side of the screen where you can access the rest of the app.

How can I switch between multiple leagues?

You can switch between all of your leagues by tapping the "My League" tab, and then tap the "Leagues" button in the upper left-hand corner. This will open up the league management utility where you can add/remove your leagues and also switch to a different league. The blue highlighted league represents your current league. To switch to a different league just tap on it!