As soon as Higgins gets hit by the coconut on the title screen, press RIGHT, LEFT, RIGHT, LEFT, A, B, A, B to access the level selection screen. Once done, press SELECT at the level selection screen to enter Test mode, which includes a sound test and other options. Note: this may only work on the original NES version.
Find the fifth skull in the first cave on the first island. Walk past the skull, then up and down to discover a hidden room. Collect the Hudson bee logo from this room, then hold LEFT and press START at the game over screen to continue game play with all prior items at the last level played. Note: this may only work on the original NES version.
Try using your hammers in a rapid fire sort of way at all times. If they disappear and make a strange noise, jump at that spot. Your character will get an egg containing a level skip, inventory item or bonus level.