I'd like to add a more detailed solution, as one of the big problems with the updated iPad app is how to actually get to the "Now Playing" screen. When listening to a song, there is a mini version of the now playing screen in the bottom left. To get to the full Now Playing screen, tap on the ALBUM ART, which will then give you access to the queue, shuffle/repeat buttons, lyrics etc. Hope this helps!
This is counter-intuitive and should be improved by Spotify.