You will access Hard Mode once you finish the game once. The enemies will be tougher and you won't be able to resume after dying.
Clear the game once to unlock Jukebox Mode.
Infinite Luiginary Attacks
Go to the Gold Shop, located at Driftwood Shore. Buy and equip two Angel Bangle items (one for each brother). With the Casual Bros. rank-up bonus in place (or with the 50% off gear and Gear Slot +1), you should now be able to use infinite Luiginary attacks.
Picking on Luigi
On the title screen, you can tap on Luigi's head to interact with him, instead of beginning your game. You can pull Luigi's mustache to tilt the game's logo, or rub his nose to make him sneeze (causing the logo to flip). Tap on his hat and the logo will instead go dark.