At the game over screen, press the following buttons to continue on the stage you were in:
Hold A and B, then press Reset. The "KINGKONG" string stands for PRG ROM checking, and the "KONAMI" string stands for CHR ROM checking. "ONE" means the test passes, "TEN" means test fails.
During the ending credits, hold down the A & B buttons at the same time right before the credits stop scrolling. If done correctly, you will get an extra message from the developers depending on how many hours it took you to beat the game. There are 24 messages in all.
At the title screen, press Right, Left, Right, Left, B, A, Up, Down, Up, Down and A. While playing the game, press on the 2nd controller the A button to skip levels or the B button to pause the game.