Once you have the Pirate Boots, return to the dungeon on Saliva Island. In the first room, dash left underneath the platform you first land on. You'll enter a mini-dungeon modelled after the original Shantae game for the Gameboy Colour. Note that you can acquire a free auto-potion by reaching the bottom of the room.
Complete the game to unlock Pirate Mode. Selecting it when starting a new game gives Shantae all pirate gear from the game's opening, allowing for sequence breaking and faster clear times.
Depending on how fast you complete the game and how many items you collect, you can unlock wallpapers for the game's title screen, which can be accessed by pressing the small star icon on the file select. Note that if you meet the conditions for multiple wallpapers in a single run, you will unlock all of them.
In order to face the game's true final boss and unlock the good ending, collect all 20 Dark Magics before entering the final boss room. This will lead to an extended final boss fight - clearing it will unlock the game's good ending.