The iPod music app on the iPad is an odd crossbreed of Apple's iTunes interface and the playback capabilities of the iPhone and iPod Touch. In some ways, it excels beyond either interface, delivering a lightening-fast version of iTunes or a more jukebox-like version of the iPod experience.
But when Apple spliced the DNA of iTunes and the iPod to create the iPad's music player, it buried one of the most useful features of both products: shuffle.
It seems a little odd that Apple would turn its back on a playback feature so popular--the company named a product after it, after all. Check out any iPod Nano or Classic, and you'll find the shuffle option right on the main menu. On the iPad, it's literally hidden.
To give Apple some credit, I don't think it's any accident that shuffle has been put on the back burner. Ever since Apple unveiled its proprietary Genius Playlist feature in 2008, and the Genius Mixes feature of 2009, they've been steering customers toward these alternatives to random shuffle (which are arguably better). To this end, Apple does include a prominent Genius playlist button on the iPad's music interface.
Genius options are nice, but if you like keeping things dumb and random, I've put together this tutorial video on how to shuffle music on the iPad.