While the iPad's Music app lists all tab options in the menu bar, the iPhone, and the iPod Touch have fewer pixels to list all. Naturally, you have four of them listed: Playlists, Artists, Songs, and Albums. This might change in some cases (Radio).
If you want to access, say, Genre or Composers, you will have to tap on More and choose the options from here. But you can change this behavior and customize the tab in Music app.
How to Customize Tabs / Menu Bar of Music App in iOS 8 or iOS 7
Step #1. Open the Music app on your iDevice.
Step #2. Tap on More from the tab/menu bar.
Step #3. Tap on Edit on the top-right.
Step #4. You can now drag and drop the ‘red' icons onto the menu bar. The colored ones are inactive (or invisible on the menu bar), and the ones that are not filled with red are the ones that are already on the menu bar.
Step #5. You can change the order of the icons by dragging from the main bunch and dropping the icon in the correct place (in the tab/menu bar.)
Step #6. Once you're done, tap on Done on the top-right.
Step #7. That's it. The menu bar should now have icons that you want to access quickly.
This is not an iOS 8-specific feature. You can follow the same procedure even in iOS 7 where you get to configure the menu bar of the Music app according to your preferences.
You'd want to put Playlists, Artists and Albums on the menu. If you are running iOS 8 and have Radio (US and Canada), you will have Radio listed first in the tab. You can configure this too – move it around to the right if you don't use Radio much.
Got any other handy tip for iOS 8's Music app? Share it in the comments!