You must first beat the first boss and then you can enter the codes. You will only be able to enter the codes once per beaten level. So if you beat two levels without using the code you will be able to use 2 times on the next level. Or you can save them and use them all in one level.

First pause the game and enter:

UP, UP, DOWN, DOWN, LEFT, RIGHT, LEFT, RIGHT, SQUARE, TRIANGLE

Now you will be at full power with double weapon.

Pause the game and enter:

UP, UP, DOWN, DOWN, LEFT, RIGHT, LEFT, RIGHT, X, CIRCLE

You will be at full power with the laser weapon.

Once you reach Stage 9, the extra mode will appear. You will now be able to play Cube Attack.

If you get hit with a bullet from a big boss during Stage 10 you will get levels from the first Gradius. If you play them once you will always have access to them.

Once you beat the game once you will unlock the Extra Edit. This can be found in the Extra Mode.

Beat the game in Extra Edit Mode to have access to more weapons.

Beat the entire game once to unlock the Boss Rush Mode.

If you beat the entire game with out continuing you will unlock the Stage Select feature.