Selecting the basic two colors is obviously done by selecting your ship with either the A or B buttons. But to get the third color, you need to select the ship by pressing C+D simultaneously.
During the ship select screen, press A to get the basic ship color, B to get the 2-Player ship color, or C+D to get the third hidden color.
At the character select screen, hold right on the joystick and button D on both the one player and two player sides, then select your character. The next screen will be a stage select menu.