When the screen says "Game Over", press Up, Up, Down, Down, Left, Right, Left, Right, Down, Up, Down, Up. Do not press any other buttons, on the final "Up" command, the game should start from where you left off.
View Final Boss
Hold Left at the title screen. This should allow you to see "Haya-Oh", the last character in the game.
Bonus Level Tip
Get to round 5 (the bonus stage) and instead of getting as many points as possible, try and hit less than three trees and weeds. If you do, at the end of the stage you'll jump off the dragon as normal and a plane will fly across the screen. Shoot it down with your gun and you'll be awarded 10,000,000 points.
Go to the sound test screen by pressing Right, Left, Down, Up at the title screen. From here, play sounds 7, 4, 3, 7, 4, 8, 1. Press Button 1 after selecting each sound. If done correctly, an optoin screen will appear that will let you turn into a JET and change the difficulty of the game.